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Chapter 7

LOCAL TESTING OF REED-MULLER CODES

From this chapter onwards, we will switch gears and talk about property testing of

codes.

7.1 Introduction

A low degree tester is a probabilistic algorithm which, given a degree parameter _ and

oracle access to a function � on arguments (which take values from some finite field � ),

has the following behavior. If � is the evaluation of a polynomial on variables with total

degree at most _ , then the low degree tester must accept with probability one. On the other

hand, if � is “far” from being the evaluation of some polynomial on variables with degree

at most _ , then the tester must reject with constant probability. The tester can query the

function � to obtain the evaluation of � at any point. However, the tester must accomplish

its task by using as few probes as possible.

Low degree testers play an important part in the construction of Probabilistically Check-

able Proofs (or PCPs). In fact, different parameters of low degree testers (for example, the

number of probes and the amount of randomness used) directly affect the parameters of the

corresponding PCPs as well as various inapproximability results obtained from such PCPs

([36, 5]). Low degree testers also form the core of the proof of MIP NEXPTIME in [9].

Blum, Luby, and Rubinfeld designed the first low degree tester, which handled the

linear case, i.e., _s G ([21]), although with a different motivation. This was followed by

a series of works that gave low degree testers that worked for larger values of the degree

parameter ([93, 42, 7]). However, these subsequent results as well as others which use low

degree testers ([9, 43]) only work when the degree is smaller than size of the field � . Alon

et al. proposed a low degree tester for any nontrivial degree parameter over the binary field�Á� [1].

A natural open problem was to give a low degree tester for all degrees for finite fields of

size between two and the degree parameter. In this chapter we (partially) solve this problem

by presenting a low degree test for multivariate polynomials over any prime field �B� .7.1.1 Connection to Coding Theory

The evaluations of polynomials in variables of degree at most _ are well known Reed-

Muller codes (note that when � ùG , we have the Reed-Solomon codes, which we con-

sidered in Chapter 6). In particular, the evaluation of polynomials in variables of degree

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at most _ over � ¯ is the Reed-Muller code or RM ¯ �?_\!` j� with parameters _ and . These

codes have length ��� and dimension � � � � � (see [28, 29, 69] for more details). Therefore,

a function has degree _ if and only if (the vector of evaluations of) the function is a valid

codeword in RM ¯ �/ ³!Z_N� . In other words, low degree testing is equivalent to locally testing

Reed-Muller codes.

7.1.2 Overview of Our Results

It is easier to define our tester over �Á� . To test if � has degree at most _ , set �a � � Z� � , and

let Í� ÷�<_úntG�� (mod 2). Pick � -vectors ;UZ�!����¥�µ!Z; ' and ; from � � � , and test ifü×½Ù | Ts � × Û � × � � 8 8 8 � × T � Ü Î Z �µ��;�n 'ü � Û Z Ü � ; � �k ; !where for notational convenience we use

; ý G (and we will stick to this convention

throughout this chapter). We remark here that a polynomial of degree at most _ always

passes the test, whereas a polynomial of degree greater than _ gets caught with non-negligible

probability � . To obtain a constant rejection probability we repeat the test �o��G�6e��� times.

The analysis of our test follows a similar general structure developed by Rubinfeld and

Sudan in [93] and borrows techniques from [93, 1]. The presence of a doubly-transitive

group suffices for the analysis given in [93]. Essentially we show that the presence of a

doubly-transitive group acting on the coordinates of the dual code does indeed allow us

to localize the test. However, this gives a weaker result. We use techniques developed in

[1] for better results, although the adoption is not immediate. In particular the interplay

between certain geometric objects described below and their polynomial representations

plays a pivotal role in getting results that are only about a quadratic factor away from

optimal query complexity.

In coding theory terminology, we show that Reed-Muller codes over prime fields are

locally testable. We further consider a new basis of Reed-Muller code over prime fields that

in general differs from the minimum weight basis. This allows us to present a novel exact

characterization of the multivariate polynomials of degree _ in variables over prime fields.

Our basis has a clean geometric structure in terms of flats [69], and unions of parallel flats

but with different weights assigned to different parallel flats1. The equivalent polynomial

and geometric representations allow us to provide an almost optimal test.

Main Result

Our results may be stated quantitatively as follows. For a given integer _ò�H��U§ILG�� and a

given real ��� ; , our testing algorithm queries � at �W. Z� n8_µ�hU Ø ]ý �X� � Z 2 points to determine

1The natural basis given in [28, 29] assigns the same weight to each parallel flat.

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whether � can be described by a polynomial of degree at most _ . If � is indeed a polynomial

of degree at most _ , our algorithm always accepts, and if � has a relative Hamming distance

at least � from every degree _ polynomial, then our algorithm rejects � with probability

at leastZ� . (In the case _]R���USIzG�� , our tester still works but more efficient testers are

known). Our result is almost optimal since any such testing algorithm must query � in at

least �s� Z� n U ] p �ý �X� � many points (see Corollary 7.5).

We extend our analysis also to obtain a self-corrector for � (as defined in [21]), in case

the function � is reasonably close to a degree _ polynomial. Specifically, we show that the

value of the function � at any given point ¬�+l���� may be obtained with good probability

by querying � on �y��U � � � random points. Using pairwise independence we can achieve

even higher probability by querying � on U ��� � � � random points and using majority logic

decoding.

7.1.3 Overview of the Analysis

The design of our tester and its analysis follows the following general paradigm first for-

malized by Rubinfeld and Sudan [93]. The analysis also uses additional ideas used in [1].

In this section, we review the main steps involved.

The first step is coming up with an exact characterization for functions that have low

degree. The characterization identifies a collection of subsets of points and a predicate

such that an input function is of low degree if and only if for every subset in the collection,

the predicate is satisfied by the evaluation of the function at the points in the subset. The

second step entails showing that the characterization is a robust characterization, that is,

the following natural tester is indeed a local tester (see section 2.3 for a formal definition):

Pick one of the subsets in the collection uniformly at random and check if the predicate is

satisfied by the evaluation of the function on the points in the chosen subset. Note that the

number of queries made by the tester is bounded above by the size of the largest subset in

the collection.

There is a natural characterization for polynomials of low degree using their alternative

interpretation as a RM code. As RM code is a linear code, a function is of low degree if

and only if it is orthogonal to every codeword in the dual of the corresponding RM code.

The problem with the above characterization is that the resulting local tester will have to

make as many queries as the maximum number of non-zero position in any dual codeword,

which can be large. To get around this problem, instead of considering all codewords in the

dual of the RM code, we consider a collection of dual codewords that have few non-zero

positions. To obtain an exact characterization, note that this collection has to generate the

dual code.

We use the well known fact that the dual of a RM code is a RM code (with different

parameters). Thus, to obtain a collection of dual codewords with low weight that generate

the dual of a RM code it is enough to find low weight codewords that generate every RM

code. To this end we show that the characteristic vector of any affine subspace (also called a

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flat in RM terminology [69]) generates certain RM codes. To complete the characterization,

we show that any RM code can be generated by flats and certain weighted characteristic

vectors of affine subspaces (which we call pseudoflats). To prove these we look at the affine

subspaces as the intersection of (a fixed number of) hyperplanes and alternatively represent

the characteristic vectors as polynomials.

To prove that the above exact characterization is robust we use the self-correcting ap-

proach ([21, 93]). Given an input � we define a related function � as follows. The value

of ����¬"� is defined to be the most frequently occurring value, or plurality, of � at correlated

random points. The major part of the analysis is to show that if � disagrees from all low

degree polynomials in a lot of places then the tester rejects with high probability.

The analysis proceeds by first showing that � and � agree on most points. Then we show

that if the tester rejects with low enough probability then � is a low degree polynomial. In

other words, if � is far enough from all low degree polynomials, then the tester rejects with

high probability. To complete the proof, we take care of the case when � is close to some

low degree polynomial separately.

7.2 Preliminaries

Throughout this chapter, we use U to denote a prime and � to denote a prime power (U Âfor some positive integer Á ) to be a prime power. In this chapter, we will mostly deal with

prime fields. We therefore restrict most definitions to the prime field setting.

For any _ò+t  k���PILG��Ñ¡ , let � denote the family of all functions over � � ¯ that are poly-

nomials of total degree at most _ (and w.l.o.g. individual degree at most �}I5G ) in vari-

ables. In particular �L+Q� if there exists coefficients @ � û � � 8 8 8 � û Ù � +Q� ¯ , for every Íy+Ê  �¡ ,?�Îä+w¢ ; !¥�¥���Y!f�JI�Ge¦ , � �Î Û Z ?�ÎäE>_ , such that�] ü� û � � 8 8 8 � û Ù � Ù Ýiý � 8 8 8 � ¯ å�Z à Ù � ý¤Ï s Ù� t � û � Ï @ � û � � 8 8 8 � û Ù � �å Î Û Z ¬ û �Î ¤ (7.1)

The codeword corresponding to a function will be the evaluation vector of � . We recall the

definition of the (Primitive) Reed-Muller code as described in [69, 29].

Definition 7.1. Let �� ÷�"�¯ be the vector space of -tuples, for Q� G , over the field � ¯ .For any � such that

; EÊ�{E÷ k���}I5G�� , the � order Reed-Muller code RM ¯ �Ñ�B!` j� is the

subspace of � ¿ � ¿¯ of all -variable polynomial functions (reduced modulo ¬ ¯Î Ig¬�Î ) of degree

at most � .This implies that the code corresponding to the family of functions - is RM ¯ �?_\!� Y� .

Therefore, a characterization for one will simply translate into a characterization for the

other.

We will be using terminology defined in Section 2.3. We now briefly review the defi-

nitions that are relevant to this chapter. For any two functions �B!��S$�� �¯ ) � ¯ , the relative

115

distance Ýc �i�"!� ���Â+v  ; !¥G�¡ between � and � is defined as Ýc �i�"!� ��� Ä û�Ì Pr � Ù |�Ù ¾  Ä�µ�/¬��[Ï ����¬"�½¡ .For a function � and a family of functions ± (defined over the same domain and range), we

say � is � - close to ± , for some; R|�gR2G , if, there exists an �~+x± , where ÝU�W�"!� ���}Ez� .

Otherwise it is � - far from ± .

A one sided testing algorithm (one-sided tester) for - is a probabilistic algorithm that

is given query access to a function � and a distance parameter � , ; R,��R�G . If �S+gJ , then

the tester should always accept � (perfect completeness), and if � is � -far from - , then

with probability at leastZ� the tester should reject � .

For vectors ¬j!Z;§+]�B�� , the dot (scalar) product of ¬ and ; , denoted ¬��^; , is defined to be� �Î Û Z ¬�Î�;XÎ , where �ÐÎ denotes the Í co-ordinate of � .

To motivate the next notation which we will use frequently, we give a definition.

Definition 7.2. For any �÷� ;, a � -flat in ���� is a � -dimensional affine subspace. Let;c Z\!������ä!�; ' +]� �� be linearly independent vectors and ;-+]� �� be a point. Then the subset�l q¢ 'ü Î Û Z ÜfÎ�;XÎ�nñ;<· �BÍ8+�  ��¡B ÜfÎä+]�C�C¦

is a � -dimensional flat. We will say that � is generated by ;CZ�!������Y!�; ' at ; . The incidence

vector of the points in a given � -flat will be referred to as the codeword corresponding to

the given � -flat.

Given a function �K$X�"�� )h�C� , for ;c Z\!������ä!Z; � !6;-+]�D�� we defineH ýÌ �<;c Z\!������ä!Z; � !6;�� Ä û�Ì ü× Û � × � � 8 8 8 � ×�� � Ù | �ý ���>;�n üÎ Ù � � � ÜfÎ�;XÎW�b! (7.2)

which is the sum of the evaluations of function � over an � -flat generated by ;�Z�!����¥�ä!Z; � ! at; . Alternatively, as we will see later in Observation 7.4, this can also be interpreted as the

dot product of the codeword corresponding to the � -flat generated by ;�Z\!������ä!Z; � at ; and that

corresponding to the function � .

While � -flats are well-known, we define a new geometric object, called a pseudoflat. A� -pseudoflat is a union of ��U.IQG�� parallel ���IQG�� -flats.

Definition 7.3. Let �ÐZb!ë �³��!������ä!f�ú��å�Z be parallel �i�oILG�� -flats ( �0�ÊG ), such that for some;q+÷� �� and all _[+%  UKIqVX ¡ , �� � Zg ½;Ân|�ä , where for any set ¶ý�O� �� and ;q+÷� �� ,;.n�¶ def ù¢�¬g n=;ú· ¬|+÷¶9¦ . We define a � -pseudoflat to be the union of the set of points�9Z to �ú��å�Z . Further, given an G (where GwEýG~E USI5 V ) and a � -pseudoflat, we define

a �i�"!ZG>� pseudoflat vector as follows. Let × � be the incidence vector of � � for �_+|  UgIqG\¡ .Then the �Ñ�"!ZG>� -pseudoflat vector is defined to be � ��å�Z� Û Z � �M × � . We will also refer to the �Ñ�B!�G£� -pseudoflat vector as a codeword.

Let � be a � -pseudoflat. Also, for �S+�  UuI5G�¡ , let � � be the �i�aI5G�� -flat generated by;c Z\!������ä!�; ' å�Z at ;"n �Ð�Z; , where ;c Z�!������Y!�; ' å�Z are linearly independent. Then we say that the

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�i�"!ZG>� -pseudoflat vector corresponding to � as well as the pseudoflat � , are generated by;"!Z;c Z\!¥�¥���ä!Z; ' å�Z at ; exponentiated along ; .

See Figure 7.1 for an illustration of the Definition 7.3.

�������������

��� �!#"-pseudoflat vector corresponding to

�! ! ! ! !�����$ $ $ $ $%%%% %

�

& & & & &

Figure 7.1: Illustration of a � -pseudoflat � defined over � �� with �g z VU!QUS : and ´ : .Picture on the left shows the points in � (recall that each of ��Z\!¥¤¥¤�¤�!f��� are G -flats or lines).

Each �µÎ (for G[E2ÍyEm� ) has UB' å�Z :points in it. The points in � are shown by filled

circles and the points in �('' [ � are shown by unfilled circles. The picture on the right is the�WVC!¥G�� -pseudoflat corresponding to � .

Given a function �K$e� �� )ù�C� , for ;c Z\!¥�¥���ä!Z; � !6;J+g� �� , for all Í3+F  U. I�VX ¡ , we define

H ÎÌ �<;c Z\!¥�¥���µ!Z; � !6;�� Ä û�Ì ü× Û � × � � 8 8 8 � ×�� � Ù | �ý Ü Î Z �X �µ��;³n ü� Ù � � � Ü � ; � �b ¤ (7.3)

As we will see later in Observation 7.5, the above can also be interpreted as the dot product

of the codeword corresponding to the ���½!ZG>� -pseudoflat vector generated by ;DZ\!������ä!Z; � at ;exponentiated along ;CZ and the codeword corresponding to the function � .

7.2.1 Facts from Finite Fields

In this section we spell out some facts from finite fields which will be used later. We begin

with a simple lemma.

Lemma 7.1. For any _9+F ¸�JI�G�¡ , � 9 Ù | ¾ @ Ï ; if and only if _k t�JIQG .Proof. First note that � 9 Ù | ¾ @ � 9 Ù | �¾ @ . Observing that for any @_+l� X¯ , @ ¯ å�Z G , it

follows that � 9 Ù | �¾ @ ¯ å�Z � 9 Ù | �¾ G- pI�GoÏ ; .

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Next we show that for all _òÏ t��I_G , � 9 Ù | �¾ @ ; . Let � be a generator of �X¯ . The sum

can be re-written as � ¯ åU�Î ÛCý � Î� N ] � ¾ �e��� å�ZN ] å�Z . The denominator is non-zero for _}Ï q�òItG and

thus, the fraction is well defined. The proof is complete by noting that � � ¯ å�Z � pG .This immediately implies the following lemma.

Lemma 7.2. Let _ëZ�!������ä!Z_ � +F »�JIQG�¡ . Thenü� × � � 8 8 8 � × � � Ù � | ¾ � � Ü �Z Ü Ø� �����bÜ �� Ï ; Ó èlCe­ Ò ©>­C�) Ó è�_fZk ±_��Ð |�����C ±_ � t�JIQG<¤ (7.4)

Proof. Note that the left hand side can be rewritten as x Î Ù � � � . � ×v�^ Ù | ¾ Ü �Î�2 ¤We will need to transform products of variables to powers of linear functions in those

variables. With this motivation, we present the following identity.

Lemma 7.3. For each �"! s.t.; RQ�§Ep��U.I�G�� there exists Ü ' +]�´X� such that

Ü ' 'å Î Û Z ¬�ÎÁ 'ü Î Û Z ��I�G�� ' å>Î ¶BÎ where ¶BÎj ü* ÚÛ ¹ á � ' � � ¿ ¹ë¿ Û Î � ü � Ù ¹ ¬ � � ' ¤ (7.5)

Proof. Consider the right hand side of the (7.5). Note that all the monomials are of degree

exactly � . Also note that x 'Î Û Z ¬CÎ appears only in the ¶ ' and nowhere else. Now consider

any other monomial of degree � that has a support of size � , where; Rü�FRH� : w.l.o.g.

assume that this monomial is à Q¬ Î �Z ¬ Î Ø� �¥���ë¬ Î S� such that Í Z"nF������n_Í � 5� . Now note that

for any ×,+Ê  �£¡ , à appears with a coefficient of � 'Î � � Î Ø � �O� � � Î S � in the expansion of � � S Ù ¹ ¬ S ��' .Further for every Í*�W� , the number of choices of ×-+   �£¡ with ·O×�·ä HÍ is exactly � ' å �' å>Î<� .Therefore, summing up the coefficients of à in the various summands ¶úÎ (along with the��I�G���' å>Î factor), we get that the coefficient of à in the right hand side of (7.5) is

þ �Í Z\!�ͽ �� !¥¤¥¤¥¤� !�Í � ÿ � 'ü Î Û � ��I�G�� ' å>Î þ �*I ���I´Í ÿ � þ �Í Z\!� Í� ��!¥¤¥¤�¤¥! Í � ÿ � ' å �ü S ÛCý ��I�G�� ' å � å S þ ��Iÿ���I ��I ÿ � þ �Í Z\!� Í� ��!¥¤¥¤�¤¥! Í � ÿ ��G�I�G�� ' å � ; ¤Moreover, it is clear that Ü ' Å� 'Z½�»Z½� �O� � �¸Z�� 5�T" (mod U ) and Ü ' Ï ; for the choice of � .

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7.3 Characterization of Low Degree Polynomials over ���In this section we present an exact characterization for the family - over prime fields.

Specifically we prove the following:

Theorem 7.1. Let _� m��UÂILG����<��nºG . (Note; E G§E U§I~V .) Let ÍÐ òU§IxVPIîG . Then a

function � belongs to � , if and only if for every ;CZ\!������ä!Z; ' � Z�!6;- +]�B�� , we haveH ÎÌ �?;c Z�!������ä!Z; ' � Z�!6;��³ ; (7.6)

As mentioned previously, a characterization for the family ( is equivalent to a char-

acterization for RM���<_\!� j� . It turns out that it is easier to characterize � when viewed as

RM���<_\!� j� . Therefore our goal is to determine whether a given word belongs to the RM

code. Since we deal with a linear code, a simple strategy will then be to check whether

the given word is orthogonal to all the codewords in the dual code. Though this yields a

characterization, this is computationally inefficient. Note however that the dot product is

linear in its input. Therefore checking orthogonality with a basis of the dual code suffices.

To make it computationally efficient, we look for a basis with small weights. The above

theorem essentially is a clever restatement of this idea.

We recall the following useful lemma which can be found in [69].

Lemma 7.4. RM ¯ �Ñ �"!� j � is a linear code with block length ��� and minimum distance �i4xnG���� È where 4 is the remainder and ö the quotient resulting from dividing �W�òIL G����� ] I,�by ���JI�G�� . Then RM ¯ �i�"!� Y � ó RM ¯ � �W�JI�G��ä�� uI~��I� G£!� Y � .

Since the dual of a RM code is again a RM code (of appropriate order), we therefore

need the generators of RM code (of arbitrary order). We first establish that flats and pseud-

oflats (of suitable dimension and exponent) indeed generate the Reed-Muller code. We then

end the section with a proof of Theorem 7.1 and a few remarks.

We begin with few simple observations about flats. Note that an � -flat � is the inter-

section of �/ KI���� hyperplanes in general position. Equivalently, it consists of all pointsÇ that satisfy �/ lIê��� linear equations over ��� (i.e., one equation for each hyperplane):�BÍu+�  ´Iï�ç¡ � �� Û Z ÜfÎ � ¬ � W;NÎ where Üë Î � !6;NÎ defines the Í hyperplane (i.e., Ç satisfies� �� Û Z Üë Î � Ç � C ; Î ). General position means that the matrix ¢X Ü\Î � ¦ has rank �^ gI���� . Note that

then the characteristic function (and by abuse of notation the incidence vector) of � can be

written as � å �å Î Û Z � G�It� �ü � Û Z ÜfÎ � ¬ � IZ;NÎ�� ��å�Z �k . G if �/Ç�Z�!����¥�ä!�Ç � �8+K�;otherwise

(7.7)

We now record a lemma here that will be used later in this section.

Lemma 7.5. For �0� � , the incidence vector of any � -flat is a linear sum of the incidence

vectors of � -flats.

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Proof. Let �� �¥n G and let Ø be an � -flat. We want to show that it is generated by a linear

combination of � flats.

Let Ø be generated by ;CZ\!������ä!Z; � å�Z�!��JZ�!������ä!�� � � Z at ; . For each non-zero vector Übε Æ�ÜfÎç Z�!¥¤�¤¥¤�!NÜ Î � � � Z � È in �?� � Z� define: Ç�Îú � � Zü � Û Z ÜfÎ � � � ¤Clearly there are ��U � � Z IÊG�� such ÇXÎ . Now for each Í[+ö  U � � Z IõG�¡ , define an � -flat ��Îgenerated by ;UZ\!������ä!Z; � å�Z�!�Ç�Î at ; . Denote the incidence vector of a flat � by G � , then we

claim that G0/õ ÷ ��UaIQG�� � V p � å�Zü Î Û Z G � ¤ (7.8)

Since the vectors ;UZ\!¥¤�¤¥¤�!Z; � å�Z\!��òZ�!¥¤¥¤¥¤�!�� � � Z are all linearly independent, we can divide the

proof in three sub cases:� Çt+yØ is of the form ;(n � � å�ZÎ Û Z ?�Î?;XÎ , for some ?eZ�!¥¤¥¤¥¤�!N? � å�Zg+|�C� : Then each flat�äÎ contributes G to the right hand side of (7.8), and therefore, the right hand side is��UaIQG��\��U � � Z I�G��³ ÷ G in �C� .� Ç[+ÞØ is of the form ;k n � � � ZÎ Û Z TeÎH�9Î for some T�Z\!¥¤¥¤�¤�!NT � � Zò+w�U� : Then the flats � �that contribute have � � @o� � � � ZÎ Û Z TeÎâ�9Î , for @0 G£!¥¤�¤¥¤�!<UuI5G . Therefore, the right

hand side of (7.8) is ��UaI�G�� � ÷ G in �C� .� ÇK+ÞØ is of the form ;8n � � å�ZÎ Û Z ?�Î�;XÎ"n � � � ZÎ Û Z TeÎH�ÐÎ : Then the flats � � that contribute

have � � @�� � � � ZÎ Û Z T<ÎH�9Î , for @y ÷ G<!�¤¥¤¥¤�!QUPI´G . Therefore, the right hand side of (7.8)

is ��UaIQG�� � ÷ G in �C� .

As mentioned previously, we give an explicit basis for RM�c �<G<!� j� . For the special case

of Ug t� , our basis coincides with the min-weight basis given in [29].2 However, in general,

our basis differs from the min-weight basis provided in [29].

The following Proposition shows that the incidence vectors of flats form a basis for the

Reed-Muller code of orders that are multiples of ��UaIQG�� .Proposition 7.6. RM�c � ��U�I[G��\ �/ òIK���b! j � is generated by the incidence vectors of the � -flats.

2The equations of the hyperplanes are slightly different in our case; nonetheless, both of them define the

same basis generated by the min-weight codewords.

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Proof. We first show that the incidence vectors of the � -flats are in RM ���N��U�I�G����^ oIL�W�b!� Y� .Recall that � is the intersection of �/ ÂI ��� independent hyperplanes. Therefore using (7.7),� can be represented by a polynomial of degree at most �^ �I ������U{IõG�� in ¬YZ\!������ä! ¬ � .Therefore the incidence vectors of � -flats are in RM�c �N��UaI�G����^ uI����b! j � .

We prove that RM���N��U[IpG��\ �/ { I ���\!� j � is generated by � -flats by induction on w I � .When 0I¥�� ; , the code consists of constants, which is clearly generated by -flats i.e.,

the whole space.

To prove for an arbitrary �/ lI ���K� ;, we show that any monomial of total degreeTKE ��UuI�G����^ SI ��� can be written as a linear sum of the incidence vectors of � -flats. Let

the monomial be ¬ û �Z �����ë¬ û þ . Rewrite the monomials as ¬úZ" �����f¬ÁZ1 203 4?eZ times

���¥��¬ÀÂ" ���¥�ë ¬C1 203 4?©Â times

. Group into

products of ��UuI5G�� (not necessarily distinct) variables as much as possible. Rewrite each

group using (7.5) with �F h��U[I|G�� . For any incomplete group of size T � , use the same

equation by setting the last ��Ug I� GòI,T>�ª � variables to the constant 1. After expansion, the

monomial can be seen to be a sum of products of at most �^ wI ��� linear terms raised to

the power of UgI�G . We can add to it a polynomial of degree at most ��UgI�G��\�/ SI��úI� G��so as to represent the resulting polynomial as a sum of polynomials, each polynomial as

in (7.7). Each such non-zero polynomial is generated by a _ flat, _P� � . By induction, the

polynomial we added is generated by �>�fnuG�� flats. Thus, by Lemma 7.5 our given monomial

is generated by � -flats.

This leads to the following observation:

Observation 7.4. Consider an � -flat generated by ;�Z\!¥�¥���ä!Z; � at ; . Denote the incidence

vector of this flat by × . Then the right hand side of (7.2) may be identified as ×y�<� , where× and � denote the vector corresponding to respective codewords and � is the dot (scalar)

product.

To generate a Reed-Muller code of any arbitrary order, we need pseudoflats. Note that

the points in a � -pseudoflat may alternatively be viewed as the space given by the union

of intersections of �/ 0I��aILG�� hyperplanes, where the union is parameterized by another

hyperplane that does not take one particular value. Concretely, it is the set of points Ç which

satisfy the following constraints over ��� :�BÍ3+F  uI~��IQG�¡ �ü � Û Z Üë Î � ¬ � ï; Î/º and

�ü � Û Z Ü � å ' � � ¬ � Ï ; � å ' ¤Thus the values taken by the points of a � -pseudoflat in its corresponding �Ñ �"!ZG>� -pseudoflat

vector is given by the polynomial� å ' å�Zå Î Û Z ��G�It� �ü � Û Z Üë Î � ¬ � IZ;NÎW� ��å�Z �ä�c � �ü � Û Z Ü � å ' � � ¬ � I�; � å ' � � (7.9)

121

Remark 7.1. Note the difference between (7.9) and the basis polynomial in [29] that (along

with the action of the affine general linear group) yields the min-weight codewords:

�j �/ ¬ÁZ\!������ä! ¬ $ �k ' å�Zå Î Û Z � G�IQ��¬CÎ"I �ÐÎW� �¥å�Z � �å� Û Z ��¬ ' I´Ë � �\!where �òZ�!����¥�ä!�� ' å�Z�!�ËúZ\!������ä!�Ë � +]�U� .

The next lemma shows that the code generated by the incidence vectors of � -flats is a

subcode of the code generated by the ���½!ZG>� -pseudoflats vectors.

Claim 7.7. The ���½!ZG>� -pseudoflats vectors, where �-�%G and GK+p  UgI�VX ¡ , generate a code

containing the incidence vectors of � -flats.

Proof. Let Ø be the incidence vector of an � -flat generated by ;�Z�!����¥�ä!Z; � at ; . Since pseud-

oflat vector corresponding to an � -pseudoflat (as well as a flat) assigns the same value to all

points in the same ����IuG�� -flat, we can describe Ø (as well as any ���½!���� -pseudoflat vector) by

giving its values on each of its U ��IyG -flats. In particular, Ø z Æ�G£!¥¤¥¤¥¤�!¥G È . Let � � be a pseud-

oflat generated by ;UZ\!������ä!Z; � exponentiated along ;CZ at ;B n �Ð�P;c Z , for each �a +]�C� , and let � �be the corresponding �>�½!�G£� - pseudoflat vector. By Definition 7.3, � � assigns a value Í�� to the�>�<I_G�� -flat generated by ;>��!������ä!�; � at ;Dn´� �k nuÍ ��; . Rewriting them in terms of the values on

its �b IuG -flats yields that � � z ÆN��U8Iø�U�% ��!���U8Iø�ÁngG�����!������ä!���U8I �ún*Í ���� !������Y!� ��U8Iø�Y IuG���� È +]� �� .

Let Ù � denote U variables for �y ; !¥G£!������Y!QU*I�G£! each taking values in ��� . Then a solution

to the following system of equationsGJ ü� Ù | ý Ù � �/ ÍjI �U� � for every

; E �äEãUaI�Gimplies that Ø � �¥å�Z� ÛCý Ù � � � , which suffices to establish the claim. Consider the identityG- z��I�G�� ü� Ù | ý � �sn´ Í � � � ��å�Z�å �which may be verified by expanding and applying Lemma 7.1. Setting Ù � to ��I�G��\��I �U� �¥å�Z�å �establishes the claim.

The next Proposition complements Proposition 7.6. Together they say that by choosing

pseudoflats appropriately, Reed-Muller codes of any given order can be generated. This

gives an equivalent representation of Reed-Muller codes. An exact characterization then

follows from this alternate representation.

Proposition 7.8. For every G|+   U{ I|VX¡ , the linear code generated by ���½!ZG>� -pseudoflat

vectors is equivalent to RM�c � ��UaIQG����^ uI����ún8G<!� Y� .

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Proof. For the forward direction, consider an � -pseudoflat � . Its corresponding �>�½!�G£� -pseudoflat vector is given by an equation similar to (7.9). Thus the codeword corresponding

to the evaluation vector of this flat can be represented by a polynomial of degree at most��UaIQG��\�/ uIZ���jnîG . This completes the forward direction.

Since monomials of degree at most ��UÐIgG��\ �/ - IF��� are generated by the incidence vectors

of � -flats, Claim 7.7 will establish the proposition for such monomials. Thus, to prove the

other direction of the proposition, we restrict our attention to monomials of degree at least��U3IÂG����^ �I��W�¥nuG and show that these monomials are generated by �>��!ZG>� -pseudoflats vectors.

Now consider any such monomial. Let the degree of the monomial be ��UäIyG����^ kI ���ënáG>� ( GPEG� �ÁE=G ). Rewrite it as in Proposition 7.6. Since the degree of the monomial is ��U.IQG����^ gI���³n>GX � , we will be left with an incomplete group of degree G£� . We make any incomplete

group complete by adding 1’s (as necessary) to the product. We then use Lemma 7.3 to

rewrite each (complete) group as a linear sum of G powered terms. After expansion, the

monomial can be seen to be a sum of product of at most �/ gI� degree ��U.IQG�� powered

linear terms and a G powered linear terms. Each such polynomial is generated either by

an �>�½!�G£� -pseudoflat vector or an � -flat. Claim 7.7 completes the proof.

The following is analogous to Observation 7.4.

Observation 7.5. Consider an � -pseudoflat, generated by ;�Z\!¥�¥���µ!Z; � at ; exponentiated

along ;UZ . Let # be its corresponding ���½!ZG>� -pseudoflat vector. Then the right hand side of

(7.3) may be interpreted as #5 �X� .

Now we prove the exact characterization.

Proof of Theorem 7.1: The proof directly follows from Lemma 7.4, Proposition 7.6,

Proposition 7.8 and Observation 7.4 and Observation 7.5. Indeed by Observation 7.4, Ob-

servation 7.5 and (7.6) are essentially tests to determine whether the dot product of the

function with every vector in the dual space of RM�c �? _\!� Y� evaluates to zero. óRemark 7.2. One can obtain an alternate characterization from Remark 7.1 which we state

here without proof.

Let _k z��U.I�G��ä�e�}nF4 (note; R�4pE÷��U. I�V> � ). Let G� îUa IF4QIFV . Let Ø �x��� with·-Ø2·U G . Define the polynomial ����¬"� Ä û�Ì x N Ù / ��¬. I���� if Ø is non-empty; and ���/ ¬��9 õG

otherwise. Then a function belong to ( if and only if for every ;UZ\!������ä!Z; ' � Z�!6;a+F�D�� , we

have ü× � Ù | ý05 / ���WÜ�ZN� ü� ×�Ø � 8 8 8 � × T p � � Ù | Tý ���>;�n ' � Zü Î Û Z ÜfÎ"�r;X Î��k ; ¤Moreover, this characterization can also be extended to certain degrees for more general

fields, i.e., �C� þ (see the next remark).

Remark 7.3. The exact characterization of low degree polynomials as claimed in [42] may

be proved using duality. Note that their proof works as long as the dual code has a min-

weight basis (see [29]). Suppose that the polynomial has degree TlE ��IQ�<6,U] I|G , then

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the dual of RM ¯ ��TD !� Y� is RM ¯ �`�W�òItG��Ñ ]IF T�ItG£!� Y� and therefore has a min-weight basis.

Note that then the dual code has min-weight �WT*npG�� . Therefore, assuming the minimum

weight codewords constitute a basis (that is, the span of all codewords with the minimum

Hamming weight is the same as the code), any T8n´G evaluations of the original polynomial

on a line are dependent and vice-versa.

7.4 A Tester for Low Degree Polynomials over � ��In this section we present and analyze a one-sided tester for ( . The analysis of the algo-

rithm roughly follows the proof structure given in [93, 1]. We emphasize that the general-

ization from [1] to our case is not straightforward. As in [93, 1] we define a self-corrected

version of the (possibly corrupted) function being tested. The straightforward adoption of

the analysis given in [93] gives reasonable bounds. However, a better bound is achieved

by following the techniques developed in [1]. In there, they show that the self-corrector

function can be interpolated with overwhelming probability. However their approach ap-

pears to use special properties of �ú� and it is not clear how to generalize their technique for

arbitrary prime fields. We give a clean formulation which relies on the flats being repre-

sented through polynomials as described earlier. In particular, Claims 7.14, 7.15 and their

generalizations appear to require our new polynomial based view.

7.4.1 Tester in ���In this subsection we describe the algorithm when underlying field is �B � .

Algorithm Test- � in �C�0. Let _k ÷ ��UaIQG��ä�e�}nF4 ,

; Et4|R¥U. I�G . Denote G� 8Ua IFVòIF4 .

1. Uniformly and independently at random select ;�Z\!������ä!Z; ' � Z�!6;- +]� �� .

2. If H �Ì �<;c Z\!¥�¥���µ!Z; ' � Z\!¤ ;\�òÏ ; , then reject, else accept.

Theorem 7.2. The algorithm Test- � in �U� is a one-sided tester for � with a success

probability at least min ��Ü<��U"' � Z �<�\! Z� � ' �76 � � T p Ø � for some constant ÜJ� ; .Corollary 7.3. Repeating the algorithm Test- - in �C� for �y� Z� T p � � n,�rU ' � times, the prob-

ability of error can be reduced to less than G�6<V .We will provide a general proof framework. However, for the ease of exposition we

prove the main technical lemmas for the case of �j � . The proof idea in the general case is

similar and the details are omitted. Therefore we will essentially prove the following.

Theorem 7.4. The algorithm Test- � in ��� is a one-sided tester for � with success prob-

ability at least min �WÜe�W�>' � Z �e�\! Z� � �76 � � ] Ó Ø p � � for some constant ÜJ� ; .

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7.4.2 Analysis of Algorithm Test- �In this subsection we analyze the algorithm described in Section 7.4.1. From Claim 7.1 it

is clear that if �K+[� , then the tester accepts. Thus, the bulk of the proof is to show that if� is � -far from � , then the tester rejects with significant probability. Our proof structure

follows that of the analysis of the test in [1]. In particular, let � be the function to be tested

for membership in � . Assume we perform Test H ÎÌ for an appropriate Í as required by the

algorithm described in Section 7.4.1. For such an Í , we define �> Îs$Á� �� ) �U� as follows:

For ;K+~ ± �� !f�Q+w�C��! denote U B � N Pr B � � 8 8 8 � B T p �  Ä�µ�?;��³IºH ÎÌ �?;� Iº;c Z\!Z;£�� !������Y!�; ' � Z�!Z;c ZN�- ÷�j ¡ .Define �X Î��<;��³ L� such that ���, Ï t�l+]�C��!<U B � N �ã U B � » with ties broken arbitrarily. With this

meaning of plurality, for all Í8 +F  U. I�VX¡h½S¢ ; ¦ , �eÎ can be written as:�X Î��?;�� plurality B � � 8 8 8 � B T p � e ���?;��µIãH ÎÌ �<;�I8;c Z\!Z;£�� !������ä!Z; ' � Z�!Z;c ZN� g ¤ (7.10)

Further we define 8 Î Ä û�Ì O�áG B � � 8 8 8 � B T p � � A   H ÎÌ �<;c Z\!¥�¥���ä!Z; ' � Z�!6;��òÏ ; ¡ (7.11)

The next lemma follows from a Markov-type argument.

Lemma 7.9. For a fixed �K$e� �� )h�U� , let �X Î�! 8 Î be defined as above. Then, Ýc �i�"!� �<Îi�9EtV 8 Î .Proof. If for some ;Â+]�"�� , Pr B � � 8 8 8 � B T p �  Ä�µ�?;��k L�µ�<;��Y I¥H ÎÌ �<;}I#;c Z\!�;£�� !����¥�ä!Z; ' � Z\!Z;c Zf�½ ¡ú��G�6<V ,then ���<;��] � �µ�<;�� . Thus, we only need to worry about the set of elements ; such that

Pr B � � 8 8 8 � B T p �  Ä�µ�<;��g ù���?;���I H ÎÌ �?;]I�;c Z\!�;£�� !����¥�ä!Z; ' � Z\!Z;c Zf�Ñ ¡oE G�6eV . If the fraction of such

elements is more than V 8 Î then that contradicts the condition that8 Î� Pr B � � 8 8 8 � B T p � � A   H ÎÌ �?;c Z\!������ä!Z; ' � Z\!6;��òÏ ; ¡ Pr B � � B Ø � 8 8 8 � B T p � � A   H ÎÌ �<;c ZµIZ;�!Z;£�� !������ä!Z; ' � Z�!6;��òÏ ; ¡ Pr B � B � � 8 8 8 � B T p �  ¸���?;��sÏ L ���?;��äI8H ÎÌ �<;*Iã;c Z\!�;£�� !����¥�ä!Z; ' � Z\!Z;c Zf�½¡i¤Therefore, we obtain ÝU�i�B!��eÎW�9ELV 8 Î .

Note that Pr B � � 8 8 8 � B T p �   �X Î��<;��Ð |���?;��³I8H ÎÌ �<;yIî;c Z�!Z;£��!������ä!Z; ' � Z�!Z;c Zf�Ñ ¡�� Z� . We now show

that this probability is actually much higher. The next lemma gives a weak bound in that

direction following the analysis in [93]. For the sake of completeness, we present the proof.

Lemma 7.10. �À;L +q� �� , I�J B � � 8 8 8 � B T p � Ù |�Ùý   �X Î��<;��. M�µ�<;���I=H ÎÌ �<;uI=;c Z\!�;£��!������ä!Z; ' � Z\!Z;c Zf�Ñ ¡��GÐIFV!U ' � Z 8 Î .

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Proof. We will use ×D!ÛÒ�!o× � !sÒ � to denote �Ñ �³nSG�� dimensional vectors over ��� . Now note that

I: �X Î��<;�� Ik ¨:9aJ^C<¨ ÓªÕ ) B � � 8 8 8 � B T p � Ù | Ùý   ü¹ Ù | T p �ý � ¹ ÚÛ<; ZÑ� ý � 8 8 8 � ý>= × ÎZ �µ�>×� Z��<;*Iã;c Zf�ún ' � Zü Û � ×ë <;X Dnî;c ZN�½¡ Ik ¨:9aJ^C<¨ ÓªÕ ) B å B � � B Ø � 8 8 8 � B T p � Ù |�Ùý   ü

¹ Ù | T p �ý � ¹ ÚÛ<; ý � 8 8 8 � ý>= �>×�Zj ntG�� Î �µ�>×�Z¥�?;*Iã;c Zf�n ' � Zü Û � ×ë <;X Dnî;��½¡ Ik ¨:9aJ^C<¨ ÓªÕ ) B � � 8 8 8 � B T p � Ù | Ùý   ü¹ Ù | T p �ý � ¹ ÚÛ<; ý � 8 8 8 � ý#= �>×� ZúntG�� Î �µ� ' � Zü Û Z ×b <;X Dn8;��Ñ ¡ (7.12)

Let ÷r %Æ?;UZ\!������ä!Z; ' � Z È and ÷ � rÆ?; � Z !������ä!Z; �' � Z È . Also we will denote Æ ; !������ä! ; È byñ . Now note thatGÐI 8 ÎäE I�J B � � 8 8 8 � B T p � � A   H ÎÌ �<;c Z\!¥�¥���ä!Z; ' � Z�!6;��k ; ¡ INJ B � � 8 8 8 � B T p � � A   ü¹ Ù | T p �ý × ÎZ �µ�>;³nZ×��©÷o�k ; ¡ INJ B � � 8 8 8 � B T p � � A  Ä�µ�>;³n8;c Zf�ún ü

¹ Ù | T p �ý � ¹ ÚÛ<; Z½� ý � 8 8 8 � ý#= × ÎZ �µ��;�n�×��©÷y�k ; ¡ I�J B � � 8 8 8 � B T p � � B  ¸���?;��jn ü

¹ Ù | T p �ý � ¹ ÚÛ<; ZÑ� ý � 8 8 8 � ý>= × ÎZ �µ�<;*I8;c Zún�×��©÷y�k ; ¡ I�J B � � 8 8 8 � B T p � � B  ¸���?;��jn ü¹ Ù | T p �ý � ¹ ÚÛ<; ý � 8 8 8 � ý>= �>×� ZúntG�� Î �µ�<;Pn�×��©÷o�³ ; ¡

Therefore for any given ×[Ï L ñ we have the following:

I�J#?<� ?  Ä�µ�?;�nZ×��©÷y�k üÔ Ù | T p �ý � Ô ÚÛ�@ I*��ÒCZj nQG�� Î �µ�<;}nZ×��©÷5n Òu�©÷ � �½ ¡ú�pG�I 8 Îand for any given Ò~ Ï L ñ ,

I�J#?<� ?  ¸�µ�<;}n� Òu�©÷ � �k ü¹ Ù | T p �ý � ¹ ÚÛ�@ I*�>×�Zj ntG�� Î �µ�<;}nZ×��r÷5n�Òu�©÷ � �½ ¡ú�|G�I 8 Î�¤

126

Combining the above two and using the union bound we get,

I�J#?<� ? BAC ü¹ Ù | T p �ý � ¹ ÚÛ�@ �>×� ZúntG�� Î ���?;}n�×��©÷y��DE ü¹ Ù | T p �ý � ¹ ÚÛ�@ üÔ Ù | T p �ý � Ô ÚÛ�@ I*�8×�ZjnQG�� Î �� ÒCZúntG�� Î �µ�<;}n�×}�©÷Ln�Òu�©÷ � � üÔ Ù | T p �ý � Ô ÚÛ�@ ��ÒCZj nQG�� Î ���?;}n Òg�©÷ � �Ñ ¡�|G�I�V���U ' � Z IQG�� 8 �|G�I�V!U ' � Z 8 Î (7.13)

The lemma now follows from the observation that the probability that the same object

is drawn from a set in two independent trials lower bounds the probability of drawing the

most likely object in one trial: Suppose the objects are ordered so that U"Î is the probability

of drawing object Í , and UúZs�=UB ���2�¥��� . Then the probability of drawing the same object

twice is � Î U �Î E � Î UÁZ�U�ÎäEã UÁZ .However, when the degree being tested is larger than the field size, we can improve the

above lemma considerably. The following lemma strengthens Lemma 7.10 when _Ð�ã UPI´Gor equivalently �Â�|G . We now focus on the �ú� case. The proof appears in Section 7.4.3.

Lemma 7.11. �À;L +q� � � , I�J B � � 8 8 8 � B T p � Ù | Ùs   �X Î��<;��. M�µ�<;���I=H ÎÌ �<;uI=;c Z\!�;£��!������ä!Z; ' � Z\!Z;c Zf�Ñ ¡��GÐIt�/�U�sntG��c� 8 Î .Lemma 7.11 will be instrumental in proving the next lemma, which shows that suffi-

ciently small

8 Î implies �X Î is the self-corrected version of the function � (the proof appears

in Section 7.4.4).

Lemma 7.12. Over �Á� , if

8 ÎPR Z� � ' � Z/� � � T p � , then the function �eÎ belongs to � (assuming�.�|G ).By combining Lemma 7.9 and Lemma 7.12 we obtain that if � is �P��G�6C�i�U�U'¥�N� -far fromÐ then

8 Î is at least �s� G�6U�Ñ �U�£'¥�N� . We next consider the case in which

8 Î is small. By Lemma

7.9, in this case, the distance Ýy |ÝU�W�"!� ��� is small. The next lemma shows that in this case

the test rejects � with probability that is close to � ' � Z Ý . This follows from the fact that in

this case, the probability over the selection of ;�Z\!������ä!Z; ' � Z\!6; , that among the �£' � Z points� Î ÜfÎ?;X ÎDn ; (where Ü�Z�!¥¤¥¤¥¤�!NÜ ' � Z- +{�"� ), the functions � and � differ in precisely one point,

is close to �£' � Z �� Ý . Observe that if they do, then the test rejects.

Lemma 7.13. Suppose; E 8 ÎòE Z� � ' � Z�� � � T p � . Let Ý denote the relative distance between� and � and

�£' � Z . Then, when ;UZ\!������ä!Z; ' � Z�!6; are chosen randomly, the probability

that for exactly one point Ç among the

points � Î ÜfÎ�;X ÎDnñ ; (where �WÜ�Z\!¥¤�¤¥¤�!NÜ ' � Zf�� +S� ' � Z� ),�µ�^ Ç��òÏ ���/ Ç�� is at least � Z� å S �Z � S �©� Ý .

127

Observe that

8 Î is at least �P�W� ' � Z Ý<� . The proof of Lemma 7.13 is deferred to Sec-

tion 7.4.5.

Proof of Theorem 7.4: Clearly if � belongs to ( , then by Claim 7.1 the tester accepts �with probability 1.

Therefore let Ýc �i�"!� �Ñ�. � � . Let T� �ÝU�i�B!� � � � , where G is as in algorithm Test- � . If

8 R Z� � ' � Z�� � � T p � then by Lemma 7.12 � � +]Рand, by Lemma 7.13,

8 Πis at least �P��� ' � Z �� TU� ,which by the definition of � is at least �P�W�>' � Z �<� . Hence

8 Î��t�§ÍÑ . Ü<���£' � Z �<�b! Z� � ' � Z�� � � T p � 2 ,

for some fixed constant ÜJ� ; . óRemark 7.4. Theorem 7.2 follows from a similar argument.

7.4.3 Proof of Lemma 7.11

Observe that the goal of Lemma 7.11 is to show that at any fixed point ; , if �£Î is interpolated

from a random hyperplane, then w.h.p. the interpolated value is the most popular vote. To

ensure this we show that if �eÎ is interpolated on two independently random hyperplanes,

then the probability that these interpolated values are the same, that is the collision prob-

ability, is large. To estimate this collision probability, we show that the difference of the

interpolation values can be rewritten as a sum of H ÎÌ on small number of random hyper-

planes. Thus if the test passes often (that is, H ÎÌ evaluates to zero w.h.p.), then this sum (by

a simple union bound) evaluates to zero often, which proves the high collision probability.

The improvement will arise because we will express differences involving H ÎÌ �� ������� as a

telescoping series to essentially reduce the number of events in the union bound. To do this

we will need the following claims. We note that a similar claim for Uw ÊV was proven by

expanding the terms on both sides in [1]. However, the latter does not give much insight

into the general case i.e., for ��� . We provide proofs that have a much cleaner structure

based on the underlying geometric structure, i.e., flats or pseudoflats.

Claim 7.14. For every �ä +w¢eVC!������Y!b�CnyGX¦ , for every ;Á�i ±;CZf�b! � !���!6;� !Z;£�� !������ä!Z; � å�Z�!Z; ��� Z�!����¥�ä!; ' � Z9+0� �� , let let ¶ �Ì �<;"! � � def ±H ýÌ �<;"!Z;£�� !������ä!Z; � å�Z�! � !Z; ��� Z\!������ä!Z; ' � Z\!¤ ;\�\¤The the following holds:¶ �Ì �?;�!��s�µIF¶ �Ì �<;"! � �k ü

û Ù | �ý eª¶ �Ì �<;}nF?*��! � �µI�¶ �Ì �?;�n�? � !��P�Egk ¤Proof. Assume ;"! � !�� are independent. If not then both sides are equal to

;and hence the

equality is trivially satisfied. To see why this claim is true for the left hand side, recall the

definition of H ýÌ �� ��� and note that the sets of points in the flat generated by ;"!Z;c �� !¥�¥���ä!Z; � å�Z\!��*!; ��� Z�!����¥�ä!Z; ' � Z at ; and the flat generated by ;�!Z;>��!¥�¥���ä!Z; � å�Z\! � !Z; ��� Z\!������ä!Z; ' � Z at ; are the

same. A similar argument works for the expression on the right hand side of the equality.

128

We claim that it is enough to prove the result for �u 2G and ;ò ÷ñ . A linear transform

(or renaming the co-ordinate system appropriately) reduces the case of �{ %G and ;0Ï ñto the case of �q G and ;´ ñ . We now show how to reduce the case of �z� G to

the �´ öG case. Fix some values Ü¥��!������ä !NÜ � å�Z\!NÜ ��� Z\!������ä !NÜ ' � Z and note that one can writeÜ�ZP ;yn, Ü\ �h;£�k n5 �����bÜ � å�Z�; � å�Zµn, Ü � �Qn, Ü ��� ZP ; ��� Z�n,Ü ' � Z%; ' � Zµn�; as Ü�ZP ;yn,Ü � �Qn�;N� , where ;f�Y � � Ù Ý ��� 8 8 8 � � å�Z½ � ��� ZÑ� 8 8 8 � ' � Z à Ü � ; � n�; . Thus,¶ �Ì �?;�!��s�k ' å�Zü� ×�Ø � 8 8 8 � ×�� �e� � ×�� p � � 8 8 8 � × T p � � Ù | ý ü� × � � ×�� � Ù | Øý �µ��Ü�ZP ;} nFÜ � �~ nñ ; � �\¤One can rewrite the other ¶ �Ì �� ��� terms similarly. Note that for a fixed vector �WÜ¥��!������Y!fÜ � å�Z�!Ü ��� Z\!������ä !NÜ ' � ZN� , the value of ;ë � is the same. Finally note that the equality in the general case

is satisfied if U"' å�Z similar equalities hold in the �� pG case.

Now consider the space F generated by ;�! � and � at 0. Note that ¶ �Ì �?;�!��s� (with;Ð 5 ñ ) is just �- �fG , where G is the incidence vector of the flat given by the equation � ; .Therefore G is equivalent to the polynomial � G�I � ��å�Z � over ��� . Similarly ¶ �Ì �<;"! � �k 5 �9�ÑG where ��� is given by the polynomial � G8IL� �¥å�Z � over ��� . We use the following polynomial

identity (in ��� )� �¥å�Z I � ��å�Z üû Ù | �ý eW � G�It�W?*�~ nî;�� �¥å�Z ¡�IL  G�IQ�W? � nî;�� �¥å�Z ¡ög (7.14)

Now observe that the polynomial � G*Iv�W?*�÷ n ;�� ��å�Z � is the incidence vector of the flat

generated by ;�Iw? å�Z � and � . Similarly, the polynomial � G9I,��? � n#;�� ��å�Z � is the incidence

vector of the flat generated by ;yIF? å�Z � and � . Therefore, interpreting the above equation

in terms of incidence vectors of flats, Observation 7.4 completes the proof assuming (7.14)

is true.

We complete the proof by proving (7.14). Consider the sum: � û Ù | �ý �W?*�Qnº;�� �¥å�Z . Ex-

panding the terms and rearranging the sums we get � �¥å�Z� ÛCý � ��å�Z� � � ��å�Z� å � ;��}� û Ù | �ý ? ��å�Z½ å � .By Lemma 7.1 the sum evaluates to �� I � �¥å�Z I ; �¥å�Z � . Similarly, � û Ù | �ý �W? � n ;�� ��å�Z �� I � ��å�Z Iã; �¥å�Z � which proves (7.14).

We will also need the following claim.

Claim 7.15. For every Í8 +{ ¢c G£!������ä!<Ua IFV�¦ , for every �ä +{ ¢eVC!������Y!ë �}ntGe¦ and for every;��i =;cZN�\! � !���!¤ ;�!�;£��!������ä!Z; � å�Z�!Z; ��� Z�!������Y!�; ' � Z9+0�D�� , denote¶ Îæ� �Ì �?;�!��s� def ±H ÎÌ �<;"!Z;£�� !������ä!Z; � å�Z�!���!�; ��� Z�!������ä!Z; ' � Z\!¤ ;\�\¤Then there exists Üë Î such that¶ Îæ� �Ì �?;�!��s�µIF¶ Îæ� �Ì �?;�! � �k tÜë Î ü

û Ù | �ý j^ ¶ Îæ� �Ì �?;}nF?*��! � �äIF¶ Îæ� �Ì �<;}nF? � !��s�Em(¤

129

Proof. As in the proof of Claim 7.14, we only need to prove the claim for �� G and;a ñ . Observe that ¶ Îæ� �Ì �?;"! � �� �S�c # � , where # � denotes the �iVC!�Í�� -pseudoflat vector

of the pseudoflat � generated by ;"! � at ; exponentiated along ; . Note that the polynomial

defining # � is just ; Î �v� ��å�Z I� G�� . Similarly we can identify the other terms with polynomials

over ��� . To complete the proof, we need to prove the following identity (which is similar

to the one in (7.14)):; Î �v� ��å�Z I � ��å�Z �k tÜfÎ üû Ù | �ý e��?;}n�?*�P� Î   G�IQ�<;�I�?*�P� ��å�Z ¡DIQ�<;}nF? � � Î  � G�It�?;*I�? � � �¥å�Z ¡ögk¤

(7.15)

where ÜëÎy ùV Î . Before we prove the identity, note that ��I�G�� �k� ��å�Z� � G in �C� . This is

because for Gg Ev� E/� , � O��I�G�����UgI, �0� . Therefore �À"ä O��I�G���� � �¥å�Z �HG� �¥å � å�Z �HG holds in �C� .

Substitution yields the desired result. Also note that � û Ù | �ý �<;�n,?*�s� Î 2I:; Î (expand and

apply Lemma 7.1). Now consider the sumüû Ù | �ý �?;}n�?*�P� Î �<;� I�?*�s� �¥å�Z ü

û Ù | �ýüý6Ï � Ï Î �ý¤Ï $ Ï ��å�Z �� I�G�� $ þ Í�>ÿ þ UaIQG� ÿ ; �¥å�Z � Îç å � å $ � � � $ ? � � $

üý¤Ï � Ï Î �ý6Ï $ Ï ��å�Z ��I�G�� $ þ Í�>ÿ þ U. I�G� ÿ ; �¥å�Z � Îç å � å $ � � � $ üû Ù | �ý ? �

� $ I:; ��å�Z � Î nL �� I�G�� � Îü � ÛCý� þ Í�£ÿ þ U. I�GUaI�G�I �>ÿ ��I�G�� �1 203 4Û Z ; Î � ��å�Z ¡ ��I�G��¥  ; Î nî; Î � �¥å�Z V Î ¡ (7.16)

Similarly one has � û Ù | �ý �<;. n5 ? � � Î �?;uI5 ? � � ��å�Z � ��I�G���  ; Î n�; Î � �¥å�Z V Î ¡ . Substituting and

simplifying one gets (7.15).

Finally, we will also need the following claim.

Claim 7.16. For every Í3 +w ¢c G£!������Y!<U.I�Vc ¦ , for every �Y +w¢eVC!������Y!6��ntGe¦ and for every;��Ñ ±; � �b! � !��� !6;� !Z;£�� !������ä!�; � å�Z\!�; ��� Z\!������ä!Z; ' � ZÐ+]� �� , there exists Üb Îä +]� X� such that¶ Îæ� �Ì �â�*!Z;��µI�¶ Îæ� �Ì � � !Z;��k üû Ù | �ý jç¶ Îæ� �Ì �?;}n�?*�*!Z;yI�?*�s�µI�¶ Îæ� �Ì �â�xn�?j;�!��QI�?j;��Nn¶ Îæ� �Ì � � n�?j;�! � Il?j;��µI�¶ ή� �Ì �<;}n�? � !�;*Il? � �nòÜfÎ jç¶ Îæ� �Ì �?;}n�?*�*! � �µIF¶ Îæ� �Ì �<;PnF? � !��P��m�m

Proof. As in the proof of Claim 7.15, the proof boils down to proving a polynomial identity

over �C� . In particular, we need to prove the following identity over �D� :� Î � G�I � ��å�Z �c I � Î � G�IF� ��å�Z �³ ÷ �v� Î I ; Î �\ � G�I � �¥å�Z ��I[� � Î ID; Î ����GÁIg� �¥å�Z �<n ; Î �â� �¥å�Z I � �¥å�Z �\¤

130

We also use that � û Ù | �ý �v�w nK?j;�� Î |I � Î and Claim 7.15 to expand the last term. Note thatÜfÎú L V Î as before.

We need one more simple fact before we can prove Lemma 7.11. For a probabil-

ity vector Ç÷ +ö  ; !¥G�¡ � , GGG ÇHGGGJI Max Î Ù � � � ¢�Ç�ÎÑ ¦Q� Max Î Ù � � � ¢�ÇX Îi¦��j�% � �Î Û Z Ç� ÎW�[ � �Î Û Z Ç�Î3�Max Î Ù � � � ¢�Ç�ν ¦� � � �Î Û Z Ç �Î GGG Ç GGG �� .Proof of Lemma 7.11: We first prove the lemma for � ý �<;�� . We fix ;´+x� �� and let Y Ä û�Ì INJ B � � 8 8 8 � B T p � Ù | Ùs   � ý �?;��3 ��µ�<;��µIîH ýÌ �?;yI8;cZ�!Z;£�� !������Y!�; ' � Z\!Z;cZf�Ñ ¡ . Recall that we want to lower

bound Y by GsIp�/�U�� npG��c� 8 ý . In that direction, we bound a slightly different but related

probability. DefineK �I�J B � � 8 8 8 � B T p � � � � � 8 8 8 � � T p � Ù | Ùs   H ýÌ �?;*Iã;cZ�!Z;£�� !������ä!Z; ' � Z\!�;cZf�³ ±H ýÌ �<;*I � Z\! � ��!������ä! � ' � Z\! � ZN�½¡Denote ÷ Æ?;UZ\!������ä!Z; ' � Z È and similarly ú . Then by the definitions of K and Y we have,Y[� K . Note that we haveK �I�J B � � 8 8 8 � B T p � � � � � 8 8 8 � � T p � Ù | Ùs   H ýÌ �<;�Iá;cZ�!Z;£��!������ä!Z; ' � Z�!Z;cZf��I{H ýÌ �?;ÁI � Z\! � �� !����¥�ä! � ' � Z�! � ZN�k ; ¡Ñ ¤

We will now use a hybrid argument. Now, for any choice of ;�Z\!������ä!Z; ' � Z and � Z\!����¥�µ!� ' � Z we have:H ýÌ �?;*I8;cZ\!�;£��!������ä!Z; ' � Z�!Z;cZf�äIãH ýÌ �?;*I � Z\! � ��!¥�¥���ä! � ' � Z\! � Zf� ±H ýÌ �?;*Iã;cZ�!Z;£�� !������Y!�; ' � Z\!Z;cZf�äI8H ýÌ �<;*Iã;cZ\!�;£�� !����¥�ä!Z; ' ! � ' � Z\!Z;cZf�nîH ýÌ �?;*I8;cZ\!�;£��!������ä!Z; ' ! � ' � Z�!Z;cZN�äI8H ýÌ �<;*I8;cZ\!Z;£�� !����¥�ä!Z; ' å�Z�! � ' ! � ' � Z�!Z;cZN�nîH ýÌ �?;*I8;cZ\!¥�¥���ä!Z; ' å�Z�! � ' ! � ' � Z\!Z;cZf�äI8H ýÌ �?;*I8;cZ\!������ä!Z; ' åU�� ! � ' å�Z\! � ' ! � ' � Z\!Z;cZf�...nîH ýÌ �?;*I8;cZ\! � �� ! � ��!������ä! � ' � Z\!Z;cZf�äI8H ýÌ �<;*I � Z\! � ��!������ä! � ' � Z\!Z;cZf�nîH ýÌ �?;*I � Zb! � �� ! � ��!������Y! � ' � Z�!Z;cZN�äI8H ýÌ �<;*I8;cZ\! � ��!������ä! � ' � Z\! � ZN�nîH ýÌ �?;*I8;cZ\! � �� ! � ��!������ä! � ' � Z\! � ZN�äI8H ýÌ �<;*I � Z\! � ��!������ä! � ' � Z\! � ZN�Consider any pair H ýÌ �<;*I8;cZ\!Z;£�� !����¥�µ!Z; � ! � ��� Z�!������Y! � ' � Z�!Z;cZN�äI8H ýÌ �<;*Iã;cZ\!�;£�� !����¥�ä!Z; � å�Z\! � � !�¥���Y! � ' � Z�!Z;cZf� that appears in the first � “rows” in the sum above. Note that H ýÌ �?;{I;cZ\!Z;£�� !����¥�ä!Z; � ! � ��� Z\!������ä! � ' � Z\!Z;cZf� and H ýÌ �<;oIî;cZ\!Z;£�� !¥�¥���ä!Z; � å�Z\! � � !������ä! � ' � Z\!Z;cZf� differ only

in a single parameter. We apply Claim 7.14 and obtain:H ýÌ �?;- I ;cZ�!Z;£�� !������Y!�; � ! � ��� Z\!¥�¥���ä! � ' � Z\!�;cZN�DI H ýÌ �<;(I ;cZ�!Z;£�� !����¥�ä!Z; � å�Z\! � � !������ä! � ' � Z\!Z;cZf�k H ýÌ �?;YIø;cZ�n ; � !Z;£�� !������ä!Z; � å�Z�! � � !������Y! � ' � Z�!Z;cZN��n H ýÌ �<;YI ;cZ� I ; � !Z;£�� !������ä!Z; � å�Zb! � � !¥�¥���µ! � ' � Zb!�;cZf�I:H ýÌ �?;B I:;cZfn � � !Z;£�� !¥�¥���ä!Z; � ! � ��� Z\!������ä! � ' � Z\!Z;cZf�ë I:H ýÌ �<;B I:; � I � � !Z;£�� !������ä!Z; � ! � ��� Z\!������Y! � ' � Z�!Z;cZN� .Recall that ; is fixed and ;>��!������ä!Z; ' � Z�! � ��!������ä! � ' � Zl+ � � � are chosen uniformly at

random, so all the parameters on the right hand side of the equation are independent and

131

uniformly distributed. Similarly one can expand the pairs H ýÌ �<;ÐI ;cZ\! � �� ! � ��!������Y! � ' � Z�!Z;cZN�UIH ýÌ �<;B I � Z\! � �� !������Y! � ' � Z�!Z;cZf� and H ýÌ �<;B I:;cZ�! � �� ! � ��!������ä! � ' � Z\! � ZN�bI:H ýÌ �?;"I � Z�! � �� !������Y! � ' � Z�! � ZN�into four H ýÌ with all parameters being independent and uniformly distributed3. Finally no-

tice that the parameters in both H ýÌ �<;B I � Z\! � ��! � ��!����¥�ä! � ' � Z�!Z;cZN� and H ýÌ �<;B I � Zb! � �� !����¥�ä! � ' � Z�!Z;cZf�are independent and uniformly distributed. Further recall that by the definition of

8 ý ,I�J � � � 8 8 8 � � T p �   H ýÌ �?G<Z\!¥�¥���ä!ZG ' � Zf�xÏ ; ¡§E 8 ý for independent and uniformly distributed G�Î s.

Thus, by the union bound, we have:I�J B � � 8 8 8 � B T p � � � � � 8 8 8 � � T p � Ù | Ùs   H ýÌ �?;cZ\!������ä!�; ' � ZN�"I H ýÌ � � Z\!¥�¥���ä! � ' � ZN�òÏ ; ¡úE÷ �/�c ��n�G ; � 8 ý ¤ (7.17)

Therefore YS� K �pG�It�^�U�PntG ; � 8 ý . A similar argument proves the lemma for �CZ¥�?;�� . The

only catch is that H Ì � � ¤�� is not symmetric– in particular in its first argument. Thus, we use

another identity as given in Claim 7.16 to resolve the issue and get four extra terms than in

the case of � ý , which results in the claimed bound of �/ �U�PntG\�U� 8 Î . óRemark 7.5. Analogously, in the case ��� we have: for every ;§+]� �� , INJ B � � B Ø � 8 8 8 � B T p � Ù |�Ùý   �X Î��<;�� L�µ�<;��µIãH ÎÌ �<;*Iã;cZ�!Z;£�� !������Y!�; ' � Z\!Z;c Zf�ún~ �µ�<;��½ ¡j �|G�I�V��N��Ua IQG�� �Pnºq���Ua I�G� �úntG� � 8 Î .The proof is similar to that of Lemma 7.11 where it can be shown K Î8 �vG- I~V��N��U§IL G��`��nq���Ua IQG� �únQG� � 8 Î , for each K Î defined for �X Î��<;�� .7.4.4 Proof of Lemma 7.12

From Theorem 7.1, it suffices to prove that if

8 Îä R Z� � ' � Z�� � � T p � then H ÎP � �<;cZ\ !¥�¥���ä!Z; ' � Z�!;��§ ;for every ;cZ�!����¥�ä!Z; ' � Z\!6;{+÷ � � � . Fix the choice of ;UZ�!������ä!Z; ' � Z�!6; . Define ÷ Æ?;cZ�!������ä!Z; ' � Z È . We will express H ÎP � �Q÷8 !¤ ;\ � as the sum of H ÎÌ �� �� � with random arguments. We

uniformly select �i�"n. G�� � random variables � Îæ� � over � � � for GPE,Í3 EL �"n. G , and GsE �. EL�"n. G .Define úä γ HÆ � Îæ�»Z�!����¥�µ! � ή� ' � Z È . We also select uniformly �i�*n5G�� random variables G�Î over�B � � for GsE,Í8EL �Ðn G . We use � Îæ� � and G¥Î ’s to set up the random arguments. Now by Lemma

7.11, for every ×_+�� ' � Z� (i.e. think of × as an ordered �i�ynqG�� -tuple over ¢ ; !¥G£!fVc ¦ ), with

probability at least G�IQ�/�U�PnQG��U� 8 Î over the choice of � ή� � and G¥Î ,�X Î��8×Á�Ð ÷§nI ;��k L�µ�>×��Ð ÷§n];��¥IøH ÎÌ �8×Á�Ð ÷§nI;<I ×Á��ú8 Z�I G<Z\!o×Á��ú³��n G� ��!������ä!o×���ú ' � Z�n G ' � Z\!o×Á��ú8Z�n G<Z �\!(7.18)

where for vectors OK!�÷v+]� ' � Z� , ÷q�rOM � ' � ZÎ Û Z ÷�Î�OyÎ , holds.

Let #òZ be the event that (7.18) holds for all ×a +]� ' � Z� . By the union bound:

Pr  ¸#òZ½ ¡j �|G�I�� ' � Z �� �/ �c�}nQG��U� 8 Î�¤ (7.19)

Assume that #òZ holds. We now need the following claims. Let Òl Æ�ÒDZ�!����¥�ä!sÒ ' � Z È be a�i�}ntG� � dimensional vector over �Á� , and denote Ò � õÆ�Ò� �� !����¥�ä!sÒ ' � Z È .3Since LNMO h ÷ n is symmetric in all but its last argument.

132

Claim 7.17. If (7.18) holds for all ×. +g � ' � Z� , then

H ýP Æ �Q÷8 !6;��k üý ÚÛ Ô Ù | Ts ` I:H ýÌ �<;cZj n ' � Zü Û � Òe � ^ �»Z\ !������ä!�; ' � Zj n ' � Zü Û � Ò< � ç � � ' � Z � !6;³n ' � Zü Û � Ò<?G¥Ñ � bn üÔ Ù | Ts ` I:H ýÌ �iV�;cZµI � ZÑ�¸Zún ' � Zü Û � Òe � ^�»Z�!������Y!ëV�; ' � Zä I � Z½� � ' � Z � n ' � Zü Û � Òe � ^� � ' � Z � !Vp;3IãG<Zj n ' � Zü Û � Ò<?G¥Ñ �nîH ýÌ � � Z½�»Zj n ' � Zü Û � Ò< � ç �¸Z\!������ä! � ZÑ� ' � Zj n ' � Zü Û � Ò< � ç � � ' � Z � !ZG<Zj n ' � Zü Û � Ò<?G¥Ñ � b(7.20)

Proof.H ýP Æ �< ÷9!6;�� ü¹ Ù | T p �s � ý �>×} �© ÷5 n�;�� ü¹ Ù | T p �s e^ I:H ýÌ �8×��© ÷Lnñ ;3IZ×}�Ãú9ZäI8G<Z\!o×��Ãú³�änîG���!������ä!o×��Ãú ' � ZjnîG ' � Z\!×}�Ãú9ZúnîG<ZN��ns�µ�>×��© ÷Lnñ ;��Ñ ¡ I ü¹ Ù | T p �s AC AC ü* ÚÛ Ô Ù | Ts �µ�8×� �r÷5 n�;�n ' � Zü Û � Òe>×���úµDn ' � Zü Û � Ò<?G¥Ñ � DEn AC üÔ ÙQP Ts � �µ�iV ×}�© ÷L n~Vp;3IZ×}�Ãú8 ZµIãG<Zj n ' � Zü Û � Ò<8×��ÃúäDn ' � Zü Û � Ò<<G�Ñ �ns�µ�>×} �Ãú8 Zj nîGeZj n ' � Zü Û � Ò<>×} �Ãúµ�n ' � Zü Û � Òe<G¥i� � b�b I üý ÚÛ Ô Ù | Ts AC ü

¹ Ù | T p �s ���8×��r÷5n�;³n ' � Zü Û � Ò<<G�Dn ' � Zü Û � Ò<>×} �Ãúµi� DEI üÔ Ù | Ts ACRAC ü¹ Ù | T p �s �µ�WV ×��© ÷L nFVp;3IZ×}�Ãú9ZµIãG<Zj n ' � Zü Û � Ò<8×��ÃúäDn ' � Zü Û � Ò<?G¥i�QDEn AC ü

¹ Ù | T p �s �µ�8×��Ãú8ZjnîG<Zún ' � Zü Û � Òe>×} �Ãúä Dn ' � Zü Û � Ò<<G¥i��DESDE

133

üý ÚÛ Ô Ù | Ts ` I:H ýÌ �?;cZY n ' � Zü Û � Ò< � ç �¸Zb!¥�¥���µ!Z; ' � Zj n ' � Zü Û � Ò< � ç � � ' � Z � !¤;�n ' � Zü Û � Ò<?G¥Ñ � bn üÔ Ù | Ts ` I:H ýÌ �WV�;cZµI � ZÑ�¸Zj n ' � Zü Û � Ò< � ç �¸Zb!¥�¥���µ!fV�; ' � ZµI � ZÑ� � ' � Z � n ' � Zü Û � Ò< � ^� � ' � Z � !Vp;3IãG<Zj n ' � Zü Û � Ò<?G¥Ñ �n H ýÌ � � ZÑ�¸ Zún ' � Zü Û � Ò< � ^�»Z\ !¥�¥���ä! � ZÑ� ' � Zj n ' � Zü Û � Ò< � ç � � ' � Z � !�GeZjn ' � Zü Û � Ò<<G�Ñ � bClaim 7.18. If (7.18) holds for all ×a +]� ' � Z� , then

H ZP � �Q÷8!6;��k üý ÚÛ Ô Ù | Ts ` I:H ZÌ �<;c Zún ' � Zü Û � Òe � ^�»Z�!������Y!�; ' � Zj n ' � Zü Û � Ò< � ^� � ' � Z � !6;³n ' � Zü Û � Ò<<G�Ñ � bn üÔ Ù | Ts ` H ZÌ �iV�;cZµI � Z½ �»Zún ' � Zü Û � Òe � ^�»Z�!������Y!ëV�; ' � Zä I � Z½� � ' � Z � n ' � Zü Û � Òe � ^� � ' � Z � !Vp;3IãG<Zj n ' � Zü Û � Ò<?G¥Ñ � b ¤ (7.21)

Proof.H ZP � �< ÷9!6;�� ü¹ Ù | T p �s ×�Z� ��Z¥�8×��© ÷5n�;�� ü¹ Ù | T p �s ×�Z e I:H ZÌ �>×}�© ÷5nñ ;3Iô×��Ãú8ZµIãG<Z�!o×} �Ãúk �ä nîG� �� !������Y!6×���ú ' � Zjn8G ' � Z�!×}�Ãú9Zjn8G<ZN�ún÷ �µ�>×} �© ÷L nñ ;��Ñ ¡ I ü¹ Ù | T p �s ×�Z ACRAC ü* ÚÛ Ô Ù | Ts �µ�>×��© ÷Lnñ ;�n ' � Zü Û � Ò<8×��ÃúäDn ' � Zü Û � Ò<<G�Ñ ��DEn AC üÔ ÙQP Ts �µ�iV ×}�© ÷L n~Vp;3IZ×}�Ãú8 ZµIãG<Zj n ' � Zü Û � Ò<8×��ÃúäDn ' � Zü Û � Ò<<G�Ñ � DERDE

I üý ÚÛ Ô Ù | Ts AC ü¹ Ù | T p �s ×�ZN�µ�8×��r÷5n�;�n ' � Zü Û � Òe<G¥Dn ' � Zü Û � Ò<8×��Ãúäi��DE

134

I üÔ Ù | Ts AC ü¹ Ù | T p �s ×� ZN�µ�iV�×��© ÷L n~Vp;3IZ×}�Ãú8 ZµIãG<Zj n ' � Zü Û � Ò<8×��ÃúäDn ' � Zü Û � Ò<?G¥Ñ � DE üý ÚÛ Ô Ù | Ts ` I:H ZÌ �<;cZj n ' � Zü Û � Ò< � ç �¸Z\!������ä!Z; ' � ZY n ' � Zü Û � Ò< � ç � � ' � Z � !6;³n ' � Zü Û � Ò<?G¥i� bn üÔ Ù | Ts ` H ZÌ �WV�;cZµI � Z½�»Zj n ' � Zü Û � Ò< � ç �¸Z\ !������ä !fV�; ' � ZµI � ZÑ� � ' � Z � n ' � Zü Û � Ò< � ç � � ' � Z � !V ;8 IãG<Zj n ' � Zü Û � Ò<<G�Ñ � b

Let #(� be the event that for every ÒÁ�B+]�Á'� !�H ÎÌ �<;cZNn � Òe � ^�»Z�!������Y!�; ' � Zfn � Ò< � ^� � ' � Z � !6;�n� Û � �}nQG�Ò<?G¥i�k ; , H ÎÌ �iV�;cZ\I � Z½�»ZNn � ' � Z Û � Òe � ^�»Z�!������Y!ëV�; ' � Zb I � Z½� ' � Zfn � ' � Z Û � Ò< � ç � � ' � Z � !fVp;� IGeZNn � ' � Z Û � Ò<?G¥Ñ �³ ; , and H ýÌ � � ZÑ�¸ ZNn � ' � Z Û � Ò< � ^�»Z\ !������ä ! � Z½� ' � ZNn � ' � Z Û � Ò< � ç � ' � Z\!ZG<Zfn � ' � Z Û � Ò<?G¥Ñ �³ ;. By the definition of

8 Î and the union bound, we have:

Pr  ¸ #( �N¡ú�pG�Il� ' � Z 8 Î�¤ (7.22)

Suppose that

8 γE Z� � ' � Z�� � � T p � holds. Then by (7.19) and (7.22), the probability that #}Zand #(� hold is strictly positive. In other words, there exists a choice of the � Îæ� � ’s and G¥Î ’sfor which all summands in either Claim 7.17 or in Claim 7.18, whichever is appropriate, is

0. This implies that H ÎP � �?;cZ\ !������ä !�; ' � Z\!6;��s ; . In other words, if

8 Î(E Z� � ' � Z/� � � T p � , then �X Îbelongs to � . óRemark 7.6. Over ��� we have: if

8 ÎoR Z� �®� �¥å�Z � ' � � �¥å�Z � � Z � � T p � , then �X Î belongs to � (if�.�|G ).In case of �U� , we can generalize (7.18) in a straightforward manner. Let #��Z denote the

event that all such events holds. We can similarly obtain

Pr  ¸# �Z ¡j�|G�I U ' � Z �X V��N�� Ua IQG�� �Pnºq��� UaI�G��úntG�� 8 ν¤ (7.23)

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Claim 7.19. Assume equivalent of (7.18) holds for all ×. +g � ' � Z� , then

H ÎP � �Q÷8 !6;�� üý ÚÛ Ô Ù | Tý ` I:H ÎÌ �<;cZj n ' � Zü Û � Ò< � ç �¸Z\!������ä!Z; ' � Zj n ' � Zü Û � Òe � ^� � ' � Z � !¤;�n ' � Zü Û � Òe<G¥i� bn üÔ Ù | Tý AC üÔ � Ù |/ý � Ô �fÚÛ Z Ò ÎZ ` I:H ÎÌ �� ÒCZP ;c ZµIt��ÒCZä IQG�� � Z½�»Zj n ' � Zü Û � Ò< � ç �¸Zb!¥�¥���µ!sÒCZP ; ' � Z\I�� ÒCZµI�G�� � Z½� � ' � Z � n ' � Zü Û � Òe � ^� � ' � Z � !ÛÒUZo;8 IQ�� ÒUZµIQG� �P G<Zún ' � Zü Û � Ò<<G¥i� bµb(7.24)

Proof.H ÎP � �< ÷9!6;�� ü¹ Ù | T p �ý × ÎZ �X Î��>×��r÷5n�;�� ü¹ Ù | T p �ý × ÎZ e^ I:H ÎÌ �8×} �© ÷5 n�;8 Iô×��Ãú8ZµIãG<Z\!6×} �Ãúk �ä n8G� �� !����¥�ä !o×��Ãú ' � ZúnîG ' � Z\!×��Ãú8ZjnîGeZf�únp���8×��© ÷Ln�;��½¡ I ü¹ Ù | T p �ý × ÎZ AC AC ü* ÚÛ Ô Ù | Tý �µ�8×��© ÷Lnñ ;�n ' � Zü Û � Òe>×} �Ãúä Dn ' � Zü Û � Ò<?G¥i� DEn AC üÔ � Ù | ý � Ô �fÚÛ Z Ò ÎZ AC üÔ Ù | Tý �µ��ÒCZ�×} �© ÷5 n ÒCZo;3 IQ�� ÒCZä IQG��� ×���ú9Zä It��ÒCZµI�G��P G<Zn ' � Zü Û � Ò<>×} �Ãúµ�n ' � Zü Û � Òe<G¥i� bµb I üý ÚÛ Ô Ù | Tý AC ü

¹ Ù | T p �ý × ÎZ �µ�>×} �© ÷L nñ ;³n ' � Zü Û � Ò<?G¥Dn ' � Zü Û � Ò<8×��Ãúä Ñ � DEI üÔ Ù | Tý AC üÔ � Ù | ý � Ô � ÚÛ Z Ò ÎZ AC ü¹ Ù | T p �ý × ÎZ �µ�� ÒCZ�×��r÷5n ÒCZo;3 IQ�� ÒCZä IQG��� ×��Ãú8 Zä It��ÒCZµIQG���G<Z

n ' � Zü Û � Ò<8×��ÃúäDn ' � Zü Û � Ò<<G�Ñ � b�b üý ÚÛ Ô Ù | Tý ` I:H ÎÌ �<;c Zún ' � Zü Û � Òe � ^�»Z�!������Y!�; ' � Zj n ' � Zü Û � Ò< � ^� � ' � Z � !6;³n ' � Zü Û � Ò<<G�Ñ � b

136

n üÔ Ù | Tý AC üÔ � Ù | ý � Ô � ÚÛ Z Ò ÎZ ` I:H ÎÌ �� ÒCZP ;cZµIQ�� ÒCZä IQG�� � ZÑ�¸Zún ' � Zü Û � Ò< � ç �¸Zb!¥�¥���ä! ÒCZP ; ' � ZI*�� ÒUZµIQG�� � Z½� � ' � Z � n ' � Zü Û � Ò< � ^� � ' � Z � !sÒCZ�;8 It��ÒCZµI�G��P GeZj n ' � Zü Û � Ò<<G�Ñ � b�b

Let #P�� be the event analogous to the event #J� in Claim 7.18. Then by the definition of

8 Î and the union bound, we have

Pr  ¸# �� ¡j�|G�I�V! U ' � Z 8 Î�¤ (7.25)

Then if we are given that

8 ÎÐR Z� �®� �¥å�Z � ' � � �¥å�Z � � Z � � T p � , then the probability that # �Z and # ��hold is strictly positive. Therefore, this implies H ÎP � �<;cZ\ !������ä!Z; ' � Z�!6;��³ ; .7.4.5 Proof of Lemma 7.13

For each 1z+g� ' � Z� , let O À be the indicator random variable whose value is 1 if and only if�µ�W1l��÷lnð;��òÏ ���W1��Z÷lnò;�� , where ÷z õÆ?;cZ�!¥¤¥¤¥¤�!Z; ' � Z È . Clearly, Pr   O À pG�¡� tÝ for every1 . It follows that the random variable O� � À O À which counts the number of points Çof the required form in which �µ�/ ÇC�PÏ ���^ Ç�� has expectation ã�   O0¡³ 5�� ' � Z Ý� �X Ý . It is not

difficult to check that the random variables O À are pairwise independent, since for any two

distinct 1-Zo ù�W1( Z½�»Z\!¥¤�¤¥¤�!f13Îæ� ' � ZN� and 1Ð�. ù�i1Ð���¸Z\!¥¤¥¤¥¤�!f1Ð��� ' � ZN� , the sums � ' � ZÎ Û Z 1( ZÑ� Î�;X Îj n ;and � ' � ZÎ Û Z 1Ð��� Î�;X Î�n�; attain each pair of distinct values in �"� � with equal probability when

the vectors are chosen randomly and independently. Since O À ’s are pairwise independent,

Var   O[¡� � À Var   O À ¡ . Since O À ’s are boolean random variables, we note

Var   O À ¡� � ã�  O �À ¡�It�Hã(   O À ¡/� � ã�  O À ¡�IQ�âã�  O À ¡/� � E¥ ã�  O À ¡i¤Thus we obtain Var   O[¡� E ã�   O[¡ , so ã(  O � ¡�E ã�  O[¡ � nñ ã(  O[¡ . Next we use the following

well known inequality which holds for a random variable O taking nonnegative, integer

values,

Pr   O � ; ¡j� �âã�  O[¡ � �ã�   O � ¡ ¤Indeed if O attains value Í with probability UD Î , then we have�âã�   O[¡/ � � � ü ÎUT ý Í�U�Î � � � ü ÎUT ý Í � U�Î � U�Î � � ER� ü ÎVT ý Í�U�Î � � ü ÎUT ý U�Î � ã(  O[¡Á� Pr   O � ; ¡Ñ !where the inequality follows by the Cauchy-Schwartz inequality. In our case, this implies

Pr   Où� ; ¡ú� �âã�   O0¡/� �ã�  O � ¡ � �âã�  O0¡/� �ã(  O[¡B nL �Hã(  O[¡ � � ã(  O[¡G8n ã�   O[¡ ¤

137

Therefore,ã�   O0¡w� Pr   O pG�¡UnFV Pr   Où�tVX¡u Pr   OM pG�¡c n~V þ ã�   O0¡G9n ã(  O[¡ I Pr   OO ÷ G\¡ ÿ Vtã�   O0¡G3n ã�  O[¡ I Pr   OM pG�¡i¤After simplification we obtain,

Pr   OM ÷ G�¡ú� G�I ã�   O[¡G8n ã(  O[¡ �*ã�   O[¡i¤The proof is complete by recalling that ã(   O[¡ä �� Ý . ó7.5 A Lower Bound and Improved Self-correction

7.5.1 A Lower Bound

The next theorem is a simple modification of a theorem in [1] and essentially implies that

our result is almost optimal.

Proposition 7.20. Let W be any family of functions � $�� �� ) �U� that corresponds to

a linear code â . Let T denote the minimum distance of the code â and let XT denote the

minimum distance of the dual code of â .

Every one-sided testing algorithm for the family W must perform �P� XTU� queries, and if the

distance parameter � is at most TU6, U � � Z , then �s� G�6X�e� is also a lower bound for the necessary

number of queries.

Lemma 7.4 and Proposition 7.20 gives us the following corollary.

Corollary 7.5. Every one-sided tester for testing ( with distance parameter � must per-

form �P� max � Z� !� ��G9nL �`�< _úntG�� mod �� UaIQG�� �N��U ] p �ý �e� �N� queries.

7.5.2 Improved Self-correction

From Lemmas 7.9, 7.11 and 7.12 the following corollary is immediate:

Corollary 7.6. Consider a function �~$�� � � ) ��� that is � -close to a degree- _ polynomial�Ê$(�D�� ) ��� , where �LR Z� � ' � Z�� � � T p � . (Assume �õ � G .) Then the function � can be

self-corrected. That is, for any given ¬x+x� �� , it is possible to obtain the value ����¬�� with

probability at least G�Il�>' � Z � by querying � on �>' � Z points on � � � .An analogous result may be obtained for the general case. We, however, improve the

above corollary slightly. The above corrector does not allow any error in the �U' � Z points it

queries. We obtain a stronger result by querying on a slightly larger flat ² , but allowing

some errors. Errors are handled by decoding the induced Reed-Muller code on ² .

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Proposition 7.21. Consider a function �K$e�"�� )h�C� that is � -close to a degree- _ polynomial��$Á�D�� ) �U� . Then the function � can be self-corrected. That is, assume� ���Ñ�onqG�� ,

then for any given ¬õ +z � �� , the value of ���/ ¬�� can be obtained with probability at leastGÐI �� Z�å �/ 8 � T p � � Ø ��U å � �8 åU� ' åc � � with U � queries to � .

Proof. Our goal is to correct the RM�c �< _\ !� Y� at the point ¬ . Assume _} ö �� UgIqG� �3 �c �onQ4 ,

where; Eõ 4HE �� UuI, V> � . Then the relative distance of the code Ý is � G- I,4} 6, U���U å ' . Note

that V! U å ' å�Z E÷ ݧE±U å ' . Recall that the local testability test requires a �Ñ��n� G�� -flat, i.e., it

tests � × � � 8 8 8 � × T p � Ù | ý Ü ��åU�Nå��Z �µ�?; ý n � ' � ZÎ Û Z ÜfÎ�;X Î��3 ; , where ;X Îä+]�D�� .

We choose a slightly larger flat, i.e., a�

-flat with� �h�i�ÂnzG�� to be chosen later.

We consider the code restricted to this�

-flat with point ¬ being the origin. We query �on this

�-flat. It is known that a majority logic decoding algorithm exists that can decode

Reed-Muller codes up to half the minimum distance for any choice of parameters (see [99]).

Thus if the number of errors is small we can recover ���/¬�� .Let the relative distance of � from the code be � and let ¶ be the set of points where it

disagrees with the closest codeword. Let the random�

-flat be ²� m¢�¬an � �Î Û Z _Ñ ÎçË�Î · _Ñ Î�+�µ! Ë�Î3+( �Â�D �� ¦ . Let the random variable ÷ ; � � 8 8 8 � � = take the value G if ¬k n � �Î Û Z Ë�Î� _Ñ Îä+K¶ and 0

otherwise. Let 7H Q� � [ ¢ ; ¦ and wq ÊÆ^ ËúZ\!������ä ! Ë1� È . Define ÷õ � ; � � 8 8 8 � � = Ù ÷ ; � � 8 8 8 � � =and

÷ �� U � IQG�� . We would like to bound the probability

Pr A� æ· ÷qIl� ·c �÷��Ý<6<VòIl�e� ¡i¤Since Pr A³  ÷� � � 8 8 8 � � pG�¡� Q� , by linearity we get ãk AР1÷} ¡� t� . Let HL z Æ? _fZ�!������ä!Z_� � È . Now

� @�G� 1÷�¡" üð Ù | � å Ýiý�à �I @�G� 1÷ ð ¡Un üð ÚÛ ð 1ø�� Ç� 1÷ ð !�÷ ð ¡ ���-Il� � �ún üð ÚÛ7Y ð 1ø�� Ç� 1÷ ð !�÷ ð ¡Un üð Û7Y ð � Z ÚÛ7Y Ù | � 1 ��ÇÁ  ÷ ð !�÷ ð ¡E ���-Il� � �ún �c �� Ua I�V> ���/�- Il� � � ���JI´� � ���� UaIQG��The above follows from the fact that when HmÏ çÙlH(� then the corresponding events ÷ ð and÷ ð are independent and therefore 1ø�� Ç� 1÷ ð !�÷ ð ¡" ; . Also, when ÷ ð and ÷ ð are dependent

then 1ø��ÇÁ   ÷ ð !�÷ ð ¡� ãk AР ÷ ð ÷ ð ¡�I ã³AР1÷ ð ¡jã³AР1÷ ð ¡j E��-Il� � .Therefore, by Chebyshev’s inequality we have (assuming ��R# U å � ' � Z � )

Pr A� æ· ÷|I´ � ·U�p��Ý<6<VòIl�e� ¡j E �U�� G�I´�<�\ �� U. I�G���WÝ<6eVsI´�<� � �

139

Now note �WÝ<6eVsIl�e�9�p�� U å ' å�Z I´�<�k ÷ � G�I´�J��U ' � Z ��U å ' å�Z . We thus have

Pr A³ ®· ÷|I´ � ·c �÷�W Ýe6<VòIl�e� ¡j E �c � G�Il�<�\ �� UaI�G����G�Il�-�hU ' � Z � � U åU� ' åU� E � U��G�Il�-�hU ' � Z � � U åU� ' åU� � nQG�� ���G�Il�-�hU ' � Z � � �hU å � �8 åU� ' åc � � ¤7.6 Bibliographics Notes

The results presented in this chapter appear in [72].

As was mentioned earlier, the study of low degree testing (along with self-correction)

dates back to the work of Blum, Luby and Rubinfeld ([21]), where an algorithm was re-

quired to test whether a given function is linear. The approach in [21] later naturally ex-

tended to yield testers for low degree polynomials over fields larger than the total degree.

Roughly, the idea is to project the given function on to a random line and then test if the

projected univariate polynomial has low degree. Specifically, for a purported degree _ func-

tion �,$��B�¯ ) � ¯ , the test works as follows. Pick vectors ; and ; from �Á�¯ (uniformly at

random), and distinct Á<Z�!������Y !,Á� � Z from � ¯ arbitrarily. Query the oracle representing � at

the _c nw G points ;D n Á�Î�; and extrapolate to a degree _ polynomial � A � B in one variable Á . Now

test for a random Á�+g��� if � A � B � ÁX �k L �µ��;�nºÁr;��(for details see [93],[42]). Similar ideas are also employed to test whether a given function

is a low degree polynomial in each of its variable (see [36, 8, 6]).

Alon et al. give a tester over field �Á� for any degree up to the number of inputs to the

function (i.e., for any non-trivial degree) [1]. In other words, their work shows that Reed-

Muller codes are locally testable. Under the coding theory interpretation, their tester picks

a random minimum-weight codeword from the dual code and checks if it is orthogonal to

the input vector. It is important to note that these minimum-weight code words generate

the Reed-Muller code.

Specifically their test works as follows: given a function �K$C¢ ; !¥Ge¦X�� ) ¢ ; !¥Ge¦ , to test if

the given function � has degree at most _ , pick �< _änLG�� -vectors ;�Z\!¥�¥���ä!Z;X � Z- +~ ¢ ; !¥Ge¦�� and

test if ü* ÚÛ4Þ á � � Z � ��� ü Î Ù Þ ;X ÎW�k ; ¤Independent of [72], Kaufman and Ron, generalizing a characterization result of [42],

gave a tester for low degree polynomials over general finite fields (see [74]). They show

that a given polynomial is of degree at most _ if and only if the restriction of the polyno-

mial to every affine subspace of suitable dimension is of degree at most _ . Following this

140

idea, their tester chooses a random affine subspace of a suitable dimension, computes the

polynomial restricted to this subspace, and verifies that the coefficients of the higher degree

terms are zero4. To obtain constant soundness, the test is repeated many times. An advan-

tage of the approach presented in this chapter is that in one round of the test (over the prime

field) we test only one linear constraint, whereas their approach needs to test multiple linear

constraints.

A basis of RM consisting of minimum-weight codewords was considered in [28, 29].

We extend their result to obtain a different exact characterization for low-degree polyno-

mials. Furthermore, it seems likely that their exact characterization can be turned into a

robust characterization following analysis similar to our robust characterization. However,

our basis is cleaner and yields a simpler analysis. We point out that for degree smaller than

the field size, the exact characterization obtained from [28, 29] coincides with [21, 93, 42].

This provides an alternate proof to the exact characterization of [42] (for more details, see

Remark 7.3 and [42]).

In an attempt to generalize our result to more general fields, we obtain an exact char-

acterization of low degree polynomials over general finite fields [71] (see [86] for more

details). This provides an alternate proof to the result of Kaufman and Ron [74] described

earlier. Specifically the result says that a given polynomial is of degree at most _ if and

only if the restriction of the polynomial to every affine subspace of dimension � � Z¯ å ¯ � � � (and

higher) is of degree at most _ .Recently Kaufman and Litsyn ([73]) show that the dual of BCH codes are locally

testable. They also give a sufficient condition for a code to be locally testable. The con-

dition roughly says that if the number of fixed length codewords in the dual of the union

of the code and its � -far coset is suitably smaller than the same in the dual of the code,

then the code is locally testable. Their argument is more combinatorial in nature and needs

the knowledge of weight-distribution of the code and thus differs from the self-correction

approach used in this work.

4Since the coefficients can be written as linear sums of the evaluations of the polynomial, this is equivalent

to check several linear constraints