Preparata codes through lattices

36 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 47, NO. 1, JANUARY 2001

Preparata Codes Through LatticesIwan M. Duursma

Abstract—We define an elementary family of lattices, fromwhich we obtain a family of extended cyclic codes with coefficientsin the modular integers. The first nontrivial subfamily is thefamily of quaternary Preparata codes. The family of dual codescoincides with the extended low-correlation sequences introducedby Kumar, Helleseth, and Calderbank.

Index Terms—Cyclic code, Goethals code, lattice, modular code,Preparata code.

I. INTRODUCTION

I N [7], Hammons, Kumar, Calderbank, Sloane, and Solé re-defined the nonlinear binary Preparata and Goethals codes

as linear codes over the quaternary alphabet .We observe that the integer lattices associated with these codeshave a straightforward and elementary description. We first de-fine and study these lattices, that are constructed with the affineline over a finite field. From the lattices, we then obtain a familyof extended cyclic codes with coefficients in the modular inte-gers. The family includes as examples the Preparata codes andthe Goethals codes.

The obtained extended cyclic codes are of lengthwith coefficients in the modular integers , for

. For , the codes coincide with extendedBose–Chaudhuri–Hocquengham (BCH) codes of lengthand designed distance. For , the codes are definedover rings of nonprime characteristic. The Preparata codesand the Goethals codes are obtained with and

, respectively. The duals of our codes are thecodes used by Kumar, Helleseth, and Calderbank [9] to obtaina new class of low-correlation sequences. Their sequencesare the natural generalization of dual BCH codes (at leastfrom one point of view, since they are able to generalize theCarlitz–Uchiyama bound to the new class of sequences). Inthis sense our codes are the duals of generalized dual BCHcodes. On the other hand, our codes do not coincide with theBCH codes over rings defined in [1], [14], [13], and [4]. Thesedefinitions lead primarily to a lower bound on the Hammingdistance. For the codes , we give a lower bound onthe Lee distance.

Reduction of the code yields codes over smalleralphabets. The coordinate-wise reductionyields the extended BCH code of designed Hamming distance

over the field .

Manuscript received January 15, 1996; revised July 28, 1999.The author was with the Gauss Research Laboratory, University of Puerto

Rico, Rio Piedras PR 00931-3334, Puerto Rico. He is now with the Departmentof Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61821USA (e-mail: [email protected]).

Communicated by V. Kumar, Asociate Editor for Coding Theory.Publisher Item Identifier S 0018-9448(01)01063-X.

II. L ATTICES

To each pair of a prime power and a positive integer weassign an integer lattice of full rank.

Definition 1: For a finite field of size , and for ,let be minimal such that . The set is the setof integer tuples such that

(1)

(2)

Lemma 1: The set is an integer lattice of full rank.It contains the sublattice .

Proof: The set is closed under addition, hencedefines a lattice. For the sublattice, (1) follows fromand (2) is obvious. The sublattice is of full rank, hence so is thelattice itself.

Lemma 2: As a subgroup of , the lattice is offinite index

Proof: The cosets of are uniquely deter-mined by an element of and by elements

.

After Theorem 9, we will prove the following. For

if and only if , where is minimal among the remaindersof modulo . The first important prop-erty of the lattice is a doubly transitive automorphism group.

Theorem 1: The lattice is invariant under coordi-nate permutations

Proof: We verify (1) and (2) for the images of the lattice.The sum is not affected by a permutation of itsterms and (2) holds. For the affine transformation, we have

By assumption, is in the lattice, and the expansionis of the form

0018–9448/01$10.00 © 2001 IEEE

DUURSMA: PREPARATA CODES THROUGH LATTICES 37

In the expansion of , we use and . Thepermutation is a field automorphism of and (1) holdswith

We consider what happens for large values of.

Proposition 1: Let and let . Then

Proof: Let . The condition

(3)

implies that, for

(4)

Hence, for . By (2) alsoThis proves that . Since both sides of(3) are th powers, we have that, for ,

. Taking th roots of both sides of (3)we obtain .

Thus at some point the result of choosinglarger is simplyblowing up a lattice by a factor. And we may assume that

. A small refinement of the same argument yieldsthefollowing proposition.

Proposition 2: Let . Then

for

Proof: We show that (4) must hold for. For this is obvious.

The numbers have a leading termin their -ary expansion. At least one of the other

digits is less than . Hence, for each there exists aand an such that . And

(4) holds for since it holds for . We conclude thatmust be constant for . The all-constant vector is indeed asolution for since

The remaining generators of can be chosen with ,for . As in the previous proof, the sublattice defined bythese generators may be described by .

The lattices , for and forwere analyzed for their density and kissing number in [6]. Inthis paper, we are primarily interested in codes obtained fromthese lattices.

III. M ODULAR CODES

With Lemma 1, we may consider the quotient space.

Definition 2: For , as in Definition 1, define thecode as

Codewords are of length with coefficients in .

Theorem 2: The code is invariant under coordinatepermutations

In particular, the code has a doubly transitive automorphismgroup.

Proof: As in Theorem 1.

For large , no new codes occur.

Proposition 3: Let and let . Then

The code is defined with alphabet . Multi-plication by yields an isomorphism

Proof: The codes and are quotientsof the same lattice (Proposition 1) by the same sublattice. In-deed, the sublattice for is determined by

And since is the smallest -multiple larger or equal than,we also have

The latter yields the alphabet size for . Proposition 1gives the isomorphism.

Lemma 3: For , the code contains theall-one word.

Proof: Conditions (1) and (2) hold for the all-one vector.

We consider properties of the codes with respect to the Leemetric. For , let the distance betweenand be definedby the absolute value . For subsets , the distanceis defined as

The distance between two cosets and is calledthe Lee distance. The Lee weight of a coset is definedas its distance to the trivial coset .

Lemma 4: For a coset , we can choose such thatthe Lee weight of is given by . This determinesuniquely in all cases but , for even. In the lattercase .

Proof: Choose with .


The Lee distance between two words of equal length overis defined as the sum of the Lee distances of corre-

sponding letters.

Theorem 3: The code has Lee distance.

Proof: With Proposition 3, the proof reduces to the case. In fact, the proposition suggests more strongly to con-

sider only codes with . Let be a nonzerocodeword. We may assume that the Lee weight at each coordi-nate is given by . This gives a division of the nonzero coor-dinates in positive and negative coordinates. For ,we make the arbitrary choice . Let

The Lee weight of the codeword equals and (1) be-comes . Replacing with

, if necessary, we may assume that . We distin-guish the cases and . In the first case

In the second case, we may assume that at least one coordinatewill be zero, or . By Proposition 2, wemay assume . Condition (1) then yields two polynomialsof equal degree that have the same leadingcoefficients. However, the polynomials are different, since theirzero sets are disjoint, and this implies that .

Lemma 5: For , we have .Proof: In general . And for , we obtain

.

While for many codes the bound is tight, several interestingcodes improve on it and it is worthwhile to investigate underwhat conditions the bound is attained.

Proposition 4: Let be a field of size . The codehas words of Lee weight (resp., )

if and only if there exists a polynomial

of degree with at least two distinct zeros (resp., of degreewith zeros distinct from ) that has all its zeros in.

Proof: The if part is straightforward and we consider theonly if part. For a given codeword, let and be de-fined as in the proof of the theorem. For a word of Lee weight

, we are in the case , say .On the other hand, , and (1)

implies . Thus,(resp., ). In the case , we may assume withProposition 2 that . Condition (1) reduces to the exis-tence of a polynomial of degree with trivial leadingcoefficients. In the case , cannot have a single zero,for that would give a lattice vector that reduces to the all-zerocodeword. In the case , we assumed that , sothat does not occur among the zeros of.

The condition on the polynomial is particularly restrictivefor close to . Recall that by definition of , . For

(resp., ), we will use the following.

Lemma 6: Let be a field of size . There exists apolynomial

of degree with at least two distinct zeros (resp., of degreewith zeros distinct from ) that has all its zeros in

if and only if (resp., and ).Proof: Observe that in both cases the polynomialhas

distinct zeros; for in both cases and have no zeros incommon. The case is obvious. For andfor any three distinct zeros of , wehave

Hence, if contains all the zeros ofit also contains all throots of unity, and is an extension of the field of ele-ments. Conversely, for an extension , we may take

.Let now be a polynomial with all its

zeros in . The substitution yields

We claim there exists a such that divides .The solutions of span a vector space over . Then

implies

for

However, is also a zero of and there exists asuch that

from which . For arbitrary , is a zero ofand, therefore, of

and . On the other hand, for a proper extensionwe can choose such that they are independentover . The linearized polynomial through has allits zeros in . A fortiori their st powers are in .

The Gray map on sends the letters to pairsof bits respectively. The image of a -code

after applying the Gray map coordinate-wise is abinary code such that the minimum Hamming dis-tance equals [7].

Theorem 4: For the codes have Leedistance

forfor oddfor odd

and size


forfor oddfor odd.

Proof: For , use Theorem 3, Proposition 4, and Lemma6. For , use Lemma 2.

The lower bounds are in fact tight. The binary images areoptimal binary codes. For , the code is just the binaryHamming code. We will establish later that the codes withand correspond with the -linear Preparata code andGoethals code, respectively, as defined in [7].

IV. REDUCED CODES

The codes are defined over . Codes oversmaller alphabets can be obtained by reduction, i.e., by applyinga map

for . By definition , and reduction yieldscodes of arbitrary alphabet size . We give explicit con-ditions for the reduced codes.

Proposition 5: Let . The reduced codeover is given by

The conditions for a vector to belong to are

(5)

(6)

Proof: The conditions for are weakened to con-ditions on only.

Theorem 5: The reduced code over is theextension of the BCH code of length and designed dis-tance . The conditions for a codeword can be written

(7)

(8)

Proof: The conditions define a linear code over andit remains to show that the conditions are those of the extendedBCH code. We may assume that the factor ispolynomial and we derive from (7)

for (9)

which is the defining condition for a narrow-sense BCH codewith designed distance. On the other hand, let a vectorsatisfy (9). We show that (7) must hold. Let

for

Then (9) with implies . And the right-hand side canbe rewritten until, after finitely many steps, we have , and(7) holds.

In general, the Lee weight drops after reducing the size ofthe alphabet. In the worst case, it drops to , the designedHamming distance of an extended BCH code. In the best case, itremains the same. Necessary conditions for a good Lee distanceafter reduction are 1) good Lee distance before reduction and 2)good shape of words of small Lee distance before reduction. Weshow that good -linear codes can in general not be expected.

Proposition 6: The reduced code has Lee dis-tance

where is the actual minimum distance of the binary cycliccode of length with zero set

for

Note that two times the designed distance of the code.

Proof: The bound follows by considering words over. For these words, the left-hand side of (5) is a

square. After taking square roots, the condition reduces to

for

But these are (7) and (8) of an extended binary BCH code, forthe word . It is now easy to find the zero set of the BCHcode. For , respectively,

respectively. In all cases,the zero set is the completion of .

As a special case, we find that the lower bound , forand odd (Theorem 4), is tight. A very curious example

that attains the upper bound was encountered by Calderbank andMcGuire [2], [3].

Proposition 7: The reduced code has Lee dis-tance , for , and otherwise.

Proof: The description of the code given here differs fromthe original. In Example 3, we show that the codes are the same.For a proof of the actual Lee distance we refer to the originalreferences. The crux is that words of Lee weight eight arise frompolynomials that factor completely over thefield of elements (see (5) in Proposition 5); or, after takingthe reciprocal, from additive polynomialsthat factor completely. Over the field of 32 elements, no suchpolynomial exists.

V. GROUP OFSYNDROMES

The codes and their reductions are linear codes. Assuch, they come with generator matrices and parity-check ma-trices. In the next section, we will derive explicit matrices from adescription of the punctured codes as cyclic codes. Before doingso, we point out what the group of syndromes looks like.

Given a parity-check matrix, let the group correspond toits column space, and the set to its columns. If thealphabet is (which can be a field, or a ring, or in general a set


of endomorphisms of ), then a codeword is a tuplewith such that

(10)

Conversely, given a group , a subset and an alphabet ,we can define a code through (10). Letbe a ring of modularintegers. In multiplicative notation, (10) becomes

(11)

The multiplicative group we are thinking of is the following.

Definition 3: For a field , let be the multiplicative groupwith elements

and let the product be defined by ordinary multiplication fol-lowed by truncation.

Propostion 8: The code is defined with alphabet, group of syndromes , and subset

.Proof: Condition (1) is defined on the component and

is a special case of the multiplicative form (11). Condition (2)is defined on the component and is a special case of theadditive form (10). In both cases, .

We will deal in the remainder of this section with the spe-cial case of -linear codes. The -linear Preparata codes andGoethals codes are defined in [7] with syndromes in the addi-tive group of a Galois ring of characteristic four. In any ringof characteristic four, we have the obvious identity

(12)

Thus, cosets of that are represented by a square form a sub-group of . For a Galois ring , the quotient is afield of characteristic two on which squaring is a bijection. Itfollows that the set of squares is a full system of representatives

for . The set is actually the Teichmüller set of representa-tives [10], [7]. Let be the set of squares, or equivalently theTeichmüller set, of the Galois ring.

Proposition 9: Let be a Galois ring of characteristic four.In particular is finite. The set gives a system of coset rep-resentatives for . For , let be the cosetrepresentative of . Elements of can be representeduniquely as , for . Addition in is given by

(13)

Proof: That the set gives a system of representativescan be found in [10], [7]. The addition law follows the generalexpression (12) but includes througha well-defined choice ofrepresentatives.

Theorem 6: Let be a Galois ring of size and identifythe field with the field of elements. The canonicalimage of in is denoted by . There exists acommutative diagram of groups as shown in the first diagramat the bottom of the page, with horizontal maps representingisomorphisms defined by the first diagram on the right and withvertical arrows given by canonical surjections.

Proof: It suffices to consider the isomorphism on top ofthe diagram. The group is generated by the elements ,for . This property does not hold for a group in general,however. The verification of the isomorphism thus reduces tosums of generators. The generator has inverse image

, for . Note that is well-defined since, for all . We have the second diagram

at the bottom of the page. The coefficient of in the image ofthe bottom map involves the following calculation:

Corollary 1: The codes for , respec-tively, are the binary Hamming code, the -linear Preparatacode, and the -linear Goethals code, respectively. The lattertwo are defined in [7].

Proof: The punctured codes are defined by agroup of syndromes respectively. Theorem 6


gives isomorphisms with respectively.The inverse image of becomes

For , the parity checks are those in the definitionof the Hamming codes as a subfield subcode. The parity checksused in [7] are

(Preparata)

(Goethals)

The difference between and does not affect the sub-ring subcode defined over . That is, the difference disappearswhen the parity-check matrix is written out in full over the ring

. Finally, as groups and are isomorphic under themap .

VI. CYCLIC CODES

We give the ideal description of the cyclic codes that are ob-tained by puncturing the codes . We also give explicitparity-check matrices. For cyclic codes with coefficients overthe modular integers , the results we need are in [4]. Inthe same paper, some classical codes are generalized by liftingthe generating polynomial. For BCH codes, the generalizationgoes back to [1], [14], and [13]. Our codes form a natural butdifferent generalization of BCH codes over a prime field.

From this point on, we will need both the additive and themultiplicative operation of Galois rings. And we recall their def-inition [10], [11], [7]. Let . For GF ,let be the lifting of its minimalpolynomial.

Definition 4: For a fixed primitive element GF ,the ring

is called theGalois ringGR of degree over .

Up to isomorphism, does not depend on the choice of. Ithas maximal ideal and

The units of the ring form a multiplicative group witha unique subgroup of order. By a slight abuse, we denote agenerator by .

Theorem 7: [4] The ring is a principalideal ring. Each ideal has a unique generator of the form

such that (14)

For a generator, set . The polynomial is determinedby the sequence

(15)

Note that .

Lemma 7: The ideal consists ofthe functions such that

The code has a parity-check matrix with columns

for

Proof: Both statements are equivalent and follow from thedefinition of the . For the parity-check matrix, we leave out theredundant conditions.

To illustrate the notation we give an example.

Example 1: Let

be the factorization of into irreducible factorsover . Let the code be the ideal of polynomials

such that

that is, . The code has aparity-check matrix with columns

for

and is generated by the polynomial , with

As a more general example, we describe the BCH codes re-ferred to at the beginning of the section.

Example 2: The BCH code of length and designeddistance with coefficients in contains codewordssuch that

It has a parity-check matrix with columns

for

The generating polynomial in standard form has

for

elsewhere.

Definition 5: For the word , we define a syn-drome

Our definition differs slightly from the sums of equal powersin [9] (notably, the role of the indexes has changed). For

, the are the usual syndromes for a cyclic code over afield, in this case . The syndromes with come intoplay for codes over rings with . In the terminology


of the Appendix, is the th phantom componentof the th coordinate of

The use of the term syndrome is justified by the following the-orem.

Theorem 8: The following are equivalent.

a) .

b) .

c) , for .

Proof:a) b): A word is in if and only if (1) holds,

which becomes Condition b) with Definition 3.b) c): An element of is the identity if and only if it is

the identity on each of the components of, for the decompo-sition given in the Appendix. The coordinate on a given com-ponent equals the identity if and only if its phantom compo-nents are trivial. The details of the latter claim are given in theAppendix.

Property c) shows how to describe the code as acyclic code. Recall that was defined through .We extend the definition of to the sequence

(16)

Theorem 9: The code contains codewordssuch that

It has a parity-check matrix with columns

for

The generating polynomial in standard form has

for

elsewhere.

Proof: Use a) c) in the previous theorem.

Example 3: The code is defined over . Itspunctured code has sequence

Therefore, columns of the parity-check matrix are of the form

For the reduced code over , the parity check reduces toand after removing the common factor two, the parity checks

define a code over . This is the code described in [3] andreferred to in Proposition 7.

The description in terms of cyclic codes allows a refinementof Lemma 2.

Corollary 2: For

if and only if , where is minimal among the remain-ders of modulo . The same conditiondetermines

Proof: By Theorem 9, the codes are determined by thesequence defined by (16). Thesequence changes atin passing from to if and only if

.

Example 4: Let . The corollary gives inequalitiesat at at at

at and at . The equalitiesoccur at .

Note the analogy and the difference with classical BCH codesover a field. For such codes, theth parity check poses the con-dition that the weightedth powersum be zero

For the codes the th parity check poses the condi-tion that the weightedth symmetricsum be zero. In character-istic , the conditions as power sum are the same forand for

. But the conditions as symmetric sum are different for. The condition as symmetric sum forcan

be translated into conditions on power sums, but only after ex-tending the characteristic by a factor(as in b) c) of Theorem8, with a proof in the Appendix).

VII. L OW-CORRELATION SEQUENCES

We briefly describe a relation with the work in [9]. FromCorollary 1, we see that the family of dual codescontains the Kerdock codes and the Delsarte–Goethals codes,which are dual to the Preparata codes and the Goethals codes,respectively. More generally, the dual codes coincide with theclass considered in [9] for the construction of low-correlationsequences. For the generating polynomials of dual cyclic codes,we have the following.

Propostion 10: Let and be dual cycliccodes of length over . The -sequences (15) of and

are related via .

Definition 6 [9]: The weighted degree of a monomial, for , equals . The weighted degree

of a polynomial is the maximum weighted degree of its mono-mials.

From Theorem 9 we see that the entries of the parity-checkmatrix of the code are the monomials of weighteddegree less than.

Definition 7 [9]: Let GR be thetrace map. For a fixed weighted degree, and for a polynomial


of weighted degree at most , the sequence is definedas

for

The sequences correspond withcodewords in , for .

Theorem 10 [9]: For a nonprincipal complex-valued char-acter on

Proof: The crucial part is an application of results by Weilthat say that the sum can be written as the sum ofalgebraic integers [15, Sec. IV, eq. 22] of absolute valueeach [15, Sec. V, eq. 26]. In [9], it is determined in the settingof Galois rings that the number of algebraic integers in the sumequals . Alternatively (observed also by Winnie Li), wecan use the isomorphism of Theorem 6 to consider the charactersum over the group instead of over a Galois ring. The group

has a well-known interpretation in Galois theory [12, Ch.V, Sec. 3, eqs. 13–16]. It is the Galois group for an extensionof the rational function field (that is, ) with conductorof degree (that is, ), which gives the correct number

.

VIII. C ONCLUSION

In the classical situation of a one-error-correcting binaryBCH code, the second condition in

is redundant. What the famous definition of the Preparatacodes by Hammons, Kumar, Calderbank, Sloane, and Solé doesis rendering it meaningful by considering the elementsem-bedded in a ring of characteristic four. What we have shownin this paper is that this construction can be realized withoutsuch an embedding by replacing the power sum conditions in theBCH code by symmetric sum conditions. This led us to a newclass of codes that includes the Preparata codes.

APPENDIX

COMPONENTS OF THEGROUP

The group (as in Definition 3) decomposes naturally, sothat the multiplication in can be replaced with group laws onthe components. Multiplication in is the same for all charac-teristics. The decomposition, however, depends on the particularcharacteristic. For a fixed characteristic, each component has thesame group law. In zero characteristic it is simple addition, innonzero characteristic it is more complicated. We establish thegroup law, and we prove that the components are isomorphic tothe additive group of a Galois ring.

For characteristic zero, the decomposition uses properties ofthe exponential function . We have

and

A general element can be written uniquely as

Multiplication on the left corresponds with coordinate-wise ad-dition on the right, yielding an isomorphism

For characteristic , we follow [5]. The function isno longer well-defined. In contrast, the function

is well defined, and can be written uniquely as

From the definition of it is clear that multiplication incan be considered coordinate-wise, that is, for eachseparately.For a fixed coordinate, can be considered a new variable.Therefore, multiplication follows the same rule at each compo-nent. We may assume . Let

We seek to describe as a function of . Both sides canbe expanded to obtain an expression of the form

Comparison of the two expressions for yields a formalequality

The can be computed with induction. They are polynomialsin with integer coefficients [5, p. 54].

Now, we want to make the relation to Galois rings. LetGR be the Galois ring of degree over , as inDefinition 4. Its group of units has a unique cyclic subgroup

of order . An element of has a uniquerepresentation

Addition is uniquely determined by the requirement that theFrobenius is a homomorphism for addition, where

Addition is determined by the addition of two elementsand with is given by


It follows that the addition law at the componentsof and the addition law for a Galois ring are thesame. From the preceding, we obtain an explicit isomorphism

, given by

We denote the latter element by . The function that assignsto the pair the element is calledthe Artin–Hasse exponential. The set of sequencesendowed with the group law from is called thegroup ofWitt-vectors relative to . The elements are thecomponentsof a Witt-vector. The elements are called thephantom com-ponentsof a Witt-vector. The phantom components have an im-portant interpretation as syndromes. The element

may be considered as a direct sum of Witt-vectors with phantomcomponents

for , and if and only ifthe phantom components are zero.

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Preparata codes through lattices