IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 38, NO. 3, MAY 1992 1101

4-Phase Sequences with Near-Optimum Correlation Properties

Serdar BoztaS, Roger Hammons, and P. Vijay Kumar

Abstract-Two families of 4-phase sequences are constructed using irreducible polynomials over Z4. Family d has period L = 2' - 1, size L +2, and maximum nontrivial correlation magnitude C,,, 5 1 + m, where r is a positive integer. Family has period L = 2(2' - l), size ( L + 2)/4, and C,,, 5 2 + m. Both families are asymptotically optimal with respect to the Welch lower bound on C,,, for complex-valued sequences. Of particular interest, Family d has the same size and period as the family of binary Gold sequences, but its maximum nontrivial correlation is smaller by a factor of 4. Since the Gold family for r odd is optimal with respect to the Welch bound restricted to binary sequences, Family d is thus superior to the best possible binary design of the same family size. Unlike the Gold design, Families d and !t? are asymptoti- cally optimal whether r is odd or even. Both families are suitable for achieving code-division multiple-access and are eas- ily implemented using shift registers. The exact distribution of correlation values is given for both families.

Index Terms-Sequence design, pseudorandom sequences, nonbinary sequences, quadriphase sequences, periodic correla- tion, code-division multiple-access.

I. INTRODUCTION

HE CODE-DIVISION multiple-access (CDMA) com- T munication system with phase-shift keying (PSK) modu- lation [ 171 provides multiple users with simultaneous access to the full communication channel bandwidth by assigning unique phase code sequences to each transmitter-receiver pair. CDMA, therefore, requires a large family of reliably distinguishable code sequences. Family size and maximum nontrivial correlation parameter C,,, are commonly used to evaluate sequence designs. Large family size is required in order to support a large number of simultaneous users. Small values of C,, are required to permit unambiguous message synchronization and to minimize interference due to compet- ing, simultaneous traffic across the channel.

In the PSK modulation format, the code symbols are the

Manuscript received April 20, 1990. This work was supported in part by the National Science Foundation under Grants NCR-87 19626 and NCR- 9016077, and by Hughes Aircraft Company under it Ph.D. fellowship program. This work was presented in part at the IEEE International Sympo- sium on Information Theory, San Diego, CA January 14- 19, 1990, and in part at the IEEE International Symposium on Information Theory, Budapest, Hungary, June 24-29, 1991.

S. Bozta~ is with the Department of Electrical and Computer Systems Engineering, Monash University, Clayton, Victoria 3 168, Australia.

R. Hammons is with the Hughes Aircraft Company, 8433 Fallbrook Avenue, Canoga Park, CA 91304-0445.

P. V. Kumar is with the Communication Sciences Institute, Electrical Engineering-Systems, University of Southern California,. Los Angeles, CA 90089-2565.

IEEE Log Number 9106948.

complex qth roots of unity. Binary PSK ( q = 2) and quadra- ture PSK ( q = 4) are the cases most often used in practice. Welch [21] and Sidelnikov [ 161 have established well-known lower bounds regarding the smallest possible C,, for a given family size M and sequence length L. In general, (see [9, introduction] for details),

cmax = [ o ( q O( m), when q = 2,

when q > 2.

Thus, the bounds suggest that non-binary designs may exist whose C,,, performance exceeds that of the best binary designs by a factor of a. In the CDMA context, this corresponds to a 3-dB improvement in signal-to-interference ratio.

In the binary case, the Gold code family with parameters L = 2' - 1 and M = L + 2, for r an odd integer, has long been known [14] to be asymptotically optimal, with respect to the Welch and Sidelnikov bounds when restricted to binary sequences. Until recently, no asymptotically optimal 4-phase family has ever been found. The comprehensive 1984 survey of 4-phase sequence designs conducted by Krone and Sarwate [8] showed none to have performance comparable to that of the binary Gold family. In this paper, two different 4-phase families are presented that are asymptotically optimal with respect to the Welch and Sidelnikov bounds. These families achieve the potential performance improvement for opti- mal nonbinary versus binary designs.

In the binary case, primitive irreducible polynomials in &[ x] yield maximal-length binary sequences with ideal autocorrelation functions. Our idea was to investigate the sequences over Z4 generated by irreducible polynomials belonging to H,[ XI. It turns out that the maximum period of a sequence generated by a degree r feedback polynomial in Z4[ x] is 2(2' - 1). Hence, a single shift register can gener- ate an entire family of cyclically distinct 4-phase sequences. Computer results indicated the existence of irreducible poly- nomials for which the correlation values of such a family are very close to the Welch lower bound. Analysis of these results led to the development of Families d and 9? pre- sented in this paper. Analogous to the role played by the finite fields GF(2') in the study of binary m-sequences, the Galois rings GR(4, r) provide a natural framework for the investigation of these maximal-length 4-phase sequences.

Family d was originally discovered by P. Sol6 [ 181, who provided all possible correlation values (but not their distribu- tion) when r is odd and, based on computer results, correctly conjectured on the possible correlation values when r is

0018-9448/92$03.00 0 1992 IEEE

1102 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 38, NO. 3, MAY 1992

even. Soli develops the theory of these sequences in terms of matrix algebras rather than Galois rings. To prove the corre- lation values, Soli first sets up an association scheme [13] whose elements are all the sequences of Family d together with their cyclic shifts. A theorem by Delsarte [6] is then invoked to show that the correlation values of the sequence family are precisely the eigenvalues of the adjacency matrices of this scheme. The proof is then completed by noting that these particular eigenvalues are computed in the paper by Liebler and Mena [ 1 13 on distance-regular digraphs. We became aware of the prior work by Sol6 only after the initial preparation of this paper. P. Udaya and M. U. Siddiqi [ 191, who were aware of SolB's work, have also independently determined the correlation distribution of Family r;9 in the case of r even. Family LJ3 is new.

In the Section 11 of this paper, the construction and correla- tion properties of the two families are summarized. In Sec- tion LII, the properties of Galois rings that are important to our derivation are surveyed. In the subsequent sections, Families d and LJ3 are discussed in detail, leading up to the determination of their distribution of correlation values.

II. MAJOR RESULTS In the construction of Families d and 93, we consider a

certain class of polynomials in Z4[ x], referred to as primi- tive basic irreducible polynomials, whose modulo 2 pro- jections are primitive, irreducible polynomials in Z2[ x ] . Specifically, let f ( x ) = x ' + ar-,x'-' + + a , x + a, be a primitive basic irreducible of degree r in Z4[x]. Consider the rth-order linear recurrence [lo, p. 4041 [20] over Z4 having characteristic polynomial2 f( x ) :

s ( t ) + a , - , s ( t - 1) + a, - , s ( t - 2) + + aos(t - r ) = 0. (2.1)

Let S ( f ) denote the set of all sequences over Z4 satisfying (2-1); S+(f) C S ( f ) , the set of all nonzero solutions; and S * ( f ) C S ( f ) , the set of solutions whose entries are not all zero-divisors. Family d is then defined to be the family S'(f) provided f ( x ) divides x2'-l - 1 in Z4[x]. Family 647 is defined to be the family S * ( f ) provided that 1) f ( x ) has order 2(2' - 1) and 2) its roots satisfy certain special restrictions (see conditions imposed on root CY in Theorem

Let 3;., j ( 7 ) = C ~ = - ~ ~ " i ( ' + ~ ) - " j ( ' ) denote the complex peri- odic correlation between the 4-phase sequences si( t + 7 ) and s,(t) of common period L where w m. By the corre- lation distribution of a family consisting of M cyclically distinct sequences of common period L, we shall mean the number of triples ( i , j , 7 ) giving rise to each complex corre-

Bozkq and Kumar [ 11, [2], [4] rediscovered Family d for odd integers r using the fact that the Galois ring GR(4, r ) is the splitting ring of certain maximal order polynomials over Z4. Hammons and Kumar [7] subsequently developed the Galois ring theoretic trace description and analysis of Family d 2 ( r odd and even) which then led to the discovery of Family 9.

The feedback polynomial commonly associated with the recurrence [22] is the reciprocal x ' f ( l / x ) of the characteristic polynomial f ( x ) . Note that, if f ( x ) is a primitive basic irreducible, so is its reciprocal. For our analysis, the characteristic polynomial is more convenient.

2).

lation {. Here, the integers i and j denote indexes between 1 and A4 inclusive, representing all cyclically distinct members of the family; and the integer 7 represents all possible cyclic shifts of the sequence and lies between 0 and L - 1 inclu- sive. The maximum nontrivial correlation parameter C,, is defined by

C,,, = max { I r i , , ( 7 ) I : either i z j or 7 # 0). (2.2)

In the following theorems, R will denote the Galois ring GR(4, r ) , T will denote the trace mapping from R onto Z4, and p will denote the (modulo 2) projection mapping from R onto the Galois field GF(2 ').

Theorem I: Family d has the following description and properties.

1) Every sequence in Family d has a unique representa- tion as s,(t) = T(y0') for some element y # 0 in R , where 0 is a fixed unit of R of multiplicative order 2' - 1. Conversely, every sequence of this form is a member of Family d.

2) The sequences of Family d are periodic with common least period L = 2' - 1. There are M = L + 2 cycli- cally distinct sequences in the family.

3) The maximum nontrivial correlation parameter of Fam- ily d is bounded above by C,, I 1 + m. The family is asymptotically optimal with respect to the Welch and Sidelnikov bounds.

4) The correlation distribution of Family d is as follows: If r = 2 s + 1, then

2 ' - 1 , - 1, - 1 + 2" + w2s,

- 1 + 2" - w2s,

- 1 - 2" + w2s,

2' + 1 times, 22' - 2 times, (2'-2 + 2 9

(2'-2 + 2 9

(2'-2 - 2"-1)

- 1 - 2 " - w2", (2'-2 - 2"-1)

-(22r - 2) times,

-(22r - 2) times,

~ ( 2 ~ ' - 2) times,

- ( 2 2 r - 2) times.

If r = 2s. then

2'- 1 , -1, 22' - 2 times, - 1 + 2",

- 1 - 2",

- 1 + w2',

- 1 - w2',

2' + 1 times,

(2'-2 + 2s-1)

(2'-2 - 2"-1)

~ ( 2 ~ ' - 2) times,

.(z2' - 2 ) times,

2'-2(22' - 2) times,

2'-2(22' - 2) times. I

BOZTAS et ai. : 4-PHASE SEQUENCES WITH NEAR-OPTIMUM CORRELATION PROPERTIES 1103

Ci, j ( 7 ) =

, let K = GF(2'), with its group of units K*. It is not hard to show that the Galois ring R is a local ring whose zero divisors form the unique maximal ideal 2 R . The units of R may be expressed [12] as a direct product of groups R* = G , x G , , where G I is a cyclic group of order 2 ' - 1 and G , is a direct product of r cyclic groups of order 2.

Obviously, the maximal ideal 2 R is the kernel of p in R , . (22-1 - 2') times, so that R / ( 2 R ) E K . In addition, there is a natural identifi-

cation of elements in K with elements in R that simplifies computations within the ideal 2 R . First, note that a polyno-

-(2"-' - 2') times, mial with only 0 and 1 as coefficients can be regarded as either an element of Z4[ x] or and element of 8,[ x ] . For any polynomial g( x ) E Uz[ x ] , let g( x ) denote the same polyno- mial viewed as an element OF Z,[x]. Then, consider the

times, 2'- 1

22'-1 - 2'-1 2(2' - l ) ,

times, - 2 , - 2 + 2" + w 2 " , (2'-2 + 2"-1)

(2'-2 + 2"-1)

. (22r-1 - 2 ') times,

- 2 + 2" - w2",

- 2 - 2" + w2", (2 '4 - 2"-1)

- 2 - 2" - &7", (2'- - 2 S - 7

. (22'- 1 - 2') \

Ci, j ( T ) = '

III. GALOIS RING PRELIMINARIES Ring Structure: The Galois ring GR(4, r ) denotes a

Galois extension of dimension r over the integer ring 8,. It turns out [12] that as in the case of finite fields, Galois rings may be constructed as quotients of the associated polynomial ring-in this case, Z 4 [ x ] . The following notation will be used. For an integer a E Z,, let Z E 8, denote its customary reduction modulo 2 . Define the polynomial reduction map-

, 2(2' - l ) , 2 r - 1 times,

- 2 , - 2 + 2"+1, (2'-2 + 2 " )

2 " ) - 2 - 2"+1, (2'-2 -

times,

2 ') times, .(22'-1 -

22'-1 - 2'-1

*(2"-' - 2') times,

2 r - 2 ( 2 2 r - 1 - 2 r ) times, 2 r - 2 ( 2 2 r - 1 - 2') times.

- 2 + w2"+1,

- 2 - w2"+1,

where f = p f. Of course, L is not a ring homomorphism, but the following relations hold for arbitrary y and v in R and their projections 7 = p(y) and V = p(v) in K:

2j. = 2 4 9 ,

2 ( y + v) = 2 4 7 + V ) ,

2 ( y v ) = 2 4 7 ; ) .

(3.1)

Note that the computations in parentheses on the left side of (3.1) occur in R , while those in parentheses on the right occur in K. Thus, arithmetic problems involving only the zero divisors of R can often be solved more conveniently in the finite field. When the context is clear, the embedding 1

will not be explicitly noted, and the element i(7) will be written as simply 7.

Automorphism Group: A basic irreducible polynomial in Z4[x] of degree r will be called a primitive basic irre- ducible if its projection by p in Z,[x] is primitive. Let

1104 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 38, NO. 3, MAY 1992

f ( x ) E Z4[ x ] be such a polynomial. Since p f ( x ) is primi- tive, it factors in K [ x ] as

where 8 is a primitive element of K. By [12], Lemma XV. 1, f( x ) has exactly, r distinct roots ao, a,, a', - e , a,- , in R , where p a i = 8".

Now consider the special case in which f ( x ) divides 1 in Z 4 [ x ] , and let /3 = ao. Since f ( x ) divides 1 and pCp = 8 is primitive in K, it follow that p has

multiplicative order 2' - 1 in R and thus generates the multiplicative subgroup G, of R*. Throughout the paper, /3 will denote a generator of G, whose projection p/3 = 8 is a primitive element in GF(2'). Since all roots of x"-l - 1 in R belong to G, = { 0, p2,,p3; - * , pZr-'}. it follows that the roots of f ( x ) are ai = p''. Now 0' - 0" is a unit whenever t # s (mod 2' - 1). Hence (see also [15]), f ( x ) factors in R [ x ] as

x z ' - l - x2'-1 -

r - 1

J = O f ( x ) = n ( x - pi). (3.3)

From the previous section, R = (1, 0, p'; * , p'- ') as a Z4-module, so that every Galois automorphism is uniquely determined by its action on 0. Since the polynomial f ( x ) is fixed by every Galois automorphism, it follows that the Galois group of R consists of exactly r different permuta- tions of the roots in (3.3). Thus, the Galois group of R is isomorphic to that of K and is generated by an automor- phism U, mapping to 0'.

Let U, denote the corresponding Galois automorphism of K that maps 8 to e', and let tr: K + Z2 denote the finite filed trace mapping. Similarly, let T : R + Z4 denote the Galois ring trace mapping defined by T ( y ) = lui(y). The following commutativity relationships between the Ga- lois ring and the corresponding Galois field are easily veri- fied:

P O U R = u K o p ,

CLOT= t r o p . (3.4)

In particular, since tr is not identically zero, it follows that the Galois ring trace is nontrivial in the sense that T ( y ) is a unit for at least one element y E R (i.e., T is onto).

Trace Distribution and Related Exponential Sums: In order to study the distribution of trace values, let each element x E R be associated with the r-tuple Ex L2 ( T ( x P 0 ) , T ( x @ ' ) ; . . , T(xP'- ' ) ) . Since R = (1, p, p'; e , O r - ' ) as a &-module and T is nontrivial, this association sets up a 1 - 1 correspondence between the elements of R and the set of all r-tuples over Z4. As a result, as x ranges over R , each of the vector components yx( i ) = T( xp') must assume each of the values 0, 1, 2, and 3 equally often. Furthermore, every r-tuple having only 0 and 2 as entries appears precisely once as Ex for some X E R \ R*. Therefore, as x ranges over R \ R*, each vector component assumes the values 0 and 2 equally often.

Based on these results, the following exponential sums are easily computed:

( - 1 p x ) = 0, X€R

IV. FAMILY d Trace Description: Let f ( x ) be a primitive basic irre-

ducible dividing x2'- - 1 in Z4[ X I . Family d is defined to be the set S ' ( f ) of nonzero sequences satisfying the linear recurrence over Z4 with characteristic polynomial f( x ) . As discussed in the previous section, R G Z4 /(f) and R = (1, 0, p'; * e , or- ' ) as a Z4-module, where 0 is a root of f( x ) in R . Furthermore, p is a generator of the cyclic group G, of R*, and f ( x ) factors over R [ x ] as shown in (3.3).

Theorem 3: For every y E R \ {0}, the sequence { sy(t)}y=o defined by s y ( t ) T(yP') is a member of Fam- ily d. Conversely, every sequence on Family d is of this form for a unique y E R \ { 0).

Proof: Direct substitution shows that all sequences of this form satisfy the linear recurrence defining Family d. A simple counting argument than shows that these sequences in fact exhaust d . 0

From the trace description, it follows that the sequences of Family d are periodic with period L = 2' - 1 equal to the multiplicative order of in R . When y is a unit, the corresponding sequences sy E d are partitioned into cycli- cally distinct equivalence classes according to the cosets yG, of G, in R*. When y # 0 is a nonunit, there is a unit v E R* such that y = 2v. In this case, using (3.1) and (3.4),

s y ( t ) = T ( y p ' ) = 2 T ( ~ p ' ) = 2p{T(vp')} = 2tr(VO'),

( 4 4

where V = pv. Therefore, the nonunits y # 0 give rise to 4-phase sequences that are cyclically equivalent, correspond- ing to all cyclic shifts of twice a binary m-sequence. Since there are 2' cosets of G, in R*, Family d thus consists of 2' + 1 cyclically distinct sequences. From (4.1), it is clear that the number of distinct versions of Family d is equal to the number of distinct binary m-sequences.

Since 0' - 6" is a unit whenever t # s(mod 2' - l ) , the set 2G1 consists precisely of the zero divisors y # 0. Hence, the cyclic equivalence classes of Family d are in 1 - 1 correspondence with the partitioning of R { 0) into subsets yoG,,ylGl,y2G1;~~,y2~G1, where yo # 0 is a nonunit and the remaining yi are units. In the following, r = {yo, y, , - * , y2,} will denote a standard set of cyclic equiva- lence class representatives in R . The corresponding sequences s i ( t ) = T(yi@') then provide a standard enu-

A

BOZTAS et al. : 4-PHASE SEQUENCES WITH NEAR-OPTIMUM CORRELATION PROPERTIES

r h ) = '

I105

I

2'- 1, - 1, - 1 + 2" + w2",

-1 + 2' - w2',

- 1 - 2' + w2',

- 1 - 2" - 02',

meration of the cyclically distinct sequences of Family d. Similarly, I'* = {y,;--, y2,} c r provides a standard set of coset representatives of G , in R*.

Correlation Properties: Let s ( t ) = T(y0' ) and s'(t) = T(y'P') be distinct sequences in Family d. The complex cross-correlation of the two sequences is given by

where y 2 y - y'. Its squared magnitude is

(4.2)

The right side of (4.2) will now be related to the exponential sum CxeR*wT(X) whose value is known from (3.5). When t # 0,l - 0' is a unit and thus may be represented as

1 - 6' = P'(1 + 2 4 , (4.3) for some integer t'(0 I t' I 2' - 2) and some z E R . In the finite field K, the Zech logarithm [5] r ( t ) is implicitly defined by the relationship = 1 - 8' . Reducing (4.3)

, modulo 2 shows that t' = ?r(t) . Note that the coset of G , containing 1 - 0' is entirely determined by the factor 1 + 22. Of course, while the element 2 2 is uniquely determined by (4.3), the element z itself is not unique.

Lemma I : In R , the element 1 - P', for 0 < t I 2' - 2, may be expressed as

I

Consequently, the cosets of G, in R* of the form (1 - Pt)G, are all distinct and cover all cosets of G, with the exception of the two cosets k GI.

Pro08 Squaring both sides of (4.3) gives 1 - 2 P + 6'' = p z r c t ) . Applying the automorphism U, to both sides gives 1 - 0'' = P2*(')(1 + 2z2) , since ~ ~ ( 2 2 ) = 22' by repeated application of (3.1) and (3.4). Combining these results yields the equation 2(P2' - P ' ) = 2z2P2"(') to be solved within the ideal 2R. By (3.1), it suffices to solve O z t + 8' = Z202*(') in K. Since one can show Z = 8'12/[1 + et / ' ] in K, it follows that z = /3'/'/[1 + Pt12] satisfies (4.3) as required.

Note that, as t varies from 1 to 2' - 2, Z = 8'12/[1 + 8 f / 2 ] assumes each value in K precisely once, except for 0 and 1 which are never assumed. Since 1 + 2 2 and 1 + 22' are in the same coset of GI in R* iff p z = p z ' , all cosets of

0 G , in R* except kG, are of the form (1 - P')G,.

r,, = {2v, - v , + v , (I - p l y , Consequently, for arbitrary v ER*, one may take

(1 - p 2 ) v ; . . , (1 - 02'-2 1.1 (4.4)

as the standard set of cyclic equivalence class representatives in R .

Theorem 4: Let r = CxeG1uT(yx). The following rela-

1) if y E R* then r lies on the circle 11 + r I 2) i f y E R \ R * a n d y # O , t h e n { = -1.

tionships hold:

= 2';

Proof: In the case of l), let y be a unit of R and consider

2'- 2 I r 12 = wT(Y(l-B')x)

t = O X € G ,

27-2 - - J(O) + w T ( Y ( l - B ' ) x ) .

xeG, t = l XEG,

By (4.4), this can be rewritten as

1 { 1 2 = (2'- 1) + U T ( x ) - r - r*; I

X€R*

so that 11 + f 1 = 2' as claimed. In the case of 2), let y # 0 be a nonunit of R . As in (4.1),

2'-2 2'- 2 J(YX) = w2T(VB') = ( - l)"(FB') = - 1,

xeG, t = O t = o

from properties of the binary trace function. 0

Since the correlations have integral real and imaginary parts, all possible values of p can be found by solving an elementary Diophantine equation. The conjugate symmetry of the correlation values together with known moments turn out to be sufficient to completely determine the weight distri- bution of Family d.

Proposition 1: For Y E R*, the only possible values of r = c ~ ~ ~ , w ~ ( Y ~ ) are as follows:

1) if r = 2s + 1, then { = - 1 k 2" k w2"; 2) if r = 2s, then { = - 1 f 2' or r = - 1 + 02'.

Proof: From the previous theorem, (1 + Re { r})' + (Im { c})' = 2'. Since Re { r } and Im { {} are integers, the desired results follow by a simple induction argument. 0

Theorem 5 (Weight Distribution): The correlation sum {(y) = C , , G , ~ T ( Y x ) assumes the following distribution of values as y varies over R:

1) If r = 2 s + 1, then

for y = 0, for 2' - 1 nonunits y # 0, for (2'-2 + 2'-')

for ( 2 ' ~ ' + 2 " ~ ' )

for(2'-' - 2'-')

for(2'-' - 2'-')

-(2' - 1) units y ,

-(2' - 1) units y,

.(2' - 1) units y ,

.(2' - 1) units y.

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 38, NO. 3, MAY 1992

[(r) = '

1 1 0 6

, 2 ) If r = 2 s , then I

2 ' - 1 , - 1 , - 1 + 2",

- 1 - 2 s ,

- 1 + w2",

- 1 - w 2 s , \

<

for y = 0, for 2' - 1 nonunits y # 0, for (2'-2 + 2 " - ' )

for ( 2 ' ~ ~ - 2 " - ' )

for 2'-2(2' - 1 ) units y ,

for ~ ' - ~ ( 2 ' - 1 ) units y .

-(2' - 1) units y,

~ ( 2 ' - 1 ) units y,

I

2 ' - 1 , - 1 , - 1 + 2" + w2",

- 1 + 2" - w2",

- 1 - 2 s + 0 2 " ,

- 1 - 2" - w2",

Proof: In either case, when y is a nonunit, the distribu- tion is clear. Hence, assume y E R * throughout the follow- ing. First consider case 1) in which r is odd. Let the standard set I'* of coset representatives of GI in R* be partitioned according to the possible values of r:

r (y ) = - 1 + 2" + w 2 " } ,

{(y) = - 1 + 2" - 0 2 " } ,

r (y) = - 1 - 2" + w 2 s } ,

{(y) = - 1 - 2 s - w 2 " } .

and - yG, are distinct for y E R * and that the corresponding values of the correlation sums are complex conjugates r and I*. As a result, one may let

note the number of cosets of each type. Since there are 2' cosets of G , inR*,

M = Ir+,+I = lI'+,-l and N = lr-,+l = IF- , - ] de-

2 ( M + N ) = 2'. (4.5)

NOW, expand the first moment sum CroRcwT(X) = 0 over the cosets of G, in R* as follows:

5 J ( X ) = c r ( d + c r(r) i = 1 xcy,G, ?,er+, + Ycr +. -

+ c r(r) + c r(r) = 0. yer-, + yer _. ~

Since r(y) is constant in each of these sums, the result is

2 ( M + N ) = 2 " + ' ( M - N ) . (4 4 Solving (4.5) and (4.6) gives M = 2'-' + 2"-' and N = 2'-2 - 2'-'. Since each coset of G, has 2' - 1 elements, the weight distribution for the case r odd is established.

Now consider case 2 ) in which r is even. The analysis now involves three unknowns:

M = I { Y E r * : r ( y ) = - 1 + 2 s } ~ ,

P = ~ { ~ ~ r * : r ( y ) = - 1 + w 2 s } ~

= I{yEr*:[(y) = - 1 - w 2 s } l

N = I{yEI '*:r(y) = -1 - 2 " } ) ,

Proceeding as before, it is easy to show that (4.5) and (4.6) are now replaced by

M + N + 2 P = 2', M + N + 2 P = 2 " ( M - N ) . (4.7)

A third equation sufficient to solve for M , N, and P can be derived by considering the following second moment:

1 1

1

2' 2'-2 2'-2 = [ i = l t = O r=O

wT(yiPt( l+P'))

- -

21-2 = ( - 1 ) T ' X ) + J - ( X )

XER * r=1 x&*

(4.8) = -2' .

When the left side of (4.8) is rewritten in terms of M , N, and P and simplified using (4.7), one obtains

M + N - 2 P = 0 .

The solution is then M = 2'-2 + 2"-' , N = 2'-2 - 2"-' , and P = 2'-2. This establishes the weight distribution for r even. U

For the correlation distribution analysis, attention is re- stricted to the cyclically distinct, standard enumeration { so(t ) , s l ( t > ; - e , s z r ( t ) } of Family d. Note that c,,,(y) =

(4.9)

~ 2 r - - 2 w s , ( l + r ) - s , ( l ) t = o = UT), where

= ? I D ' - yJ9

7, 3 7, E r . (4.10)

Thus, the crux of the problem lies in determining for each y E R the number of triples ( i , j , 7 ) satisfying (4.10).

Theorem 6 (Correlation Distribution): For Family d , the correlation distribution is as follows

1) If r = 2 s + 1, then,

2' + 1 times, 22' - 2 times, (2'-2 + 2"-1)

(2T-2 + 2 9

(2'-* - 2s -1 )

(27-2 - 2"-1)

. ( 2 2 r - 2 ) times,

.(2" - 2 ) times,

. ( 2 2 r - 2 ) times,

(z2' - 2 ) times.

BOZTAS et al. : 4-PHASE SEQUENCES WITH NEAR-OPTIMUM CORRELATION PROPERTIES 1107

2) If r = 2s , then

2 ' - 1 , -1 , 2'' - 2 times, -1 + 2",

, - 1 - 2",

- 1 + 02", - 1 - w 2 " ,

2' + 1 times,

(2'-2 + 2"-1)

( 2 ' 4 - 2s-1) .(2" - 2 ) times,

.(2" - 2 ) times, 2 ' ~ ~ ( 2 ~ ' - 2 ) times,

2'-2(22' - 2 ) times.

Pro08 Consider (4.10) with y E R fixed. In any solu- tion, the parameters i and 7 are uniquely determined by the choice of yj . Hence the number of triples ( i , j , 7) satisfying (4.10) is equal to the number of possible choices for y j from r. The special case y = 0 permits a solution for each choice of y j , so there is a total of 2' + 1 triples in this case. When y # 0, (4.10) has a solution iff y # -7,. In this case, note that by (4.4) the set { -yo, - yl; - e , - y Z r } is also a valid set of distinct cyclic equivalence class representatives in R. Thus, if y is not one of the equivalence class representatives { -yi};Lo, all 2' + 1 choices of y j in yield solutions. If y is such a representative, only 2' of the y , ~ r are possible. Hence, within each cyclic equivalence class in R , there are 2 ' - 2 members having 2 + 1 solutions each and one mem- ber having 2' solutions. Thus, each cyclic equivalence class in R provides a total of (2' + 1)(2' - 2) + 2' = 4' - 2 triples (i, j , 7 ) satisfying (4.10). The correlation distribution then follows directly from the weight distribution results (dividing by 2' - 1 and multiplying by 4' - 2 as appropri-

In-phase and Quadrature Sequences: In the engineering literature, it is customary to regard a 4-phase sequence s ( t ) as the composition of two binary sequences u ( t ) and u ( t ) referred to as the in-phase and quadrature components, re- spectively. From [8, (4)],

ate).

On the other hand, every 4-phase sequence in Family d may be uniquely written as

s ( t ) = 2 a ( t ) + b ( t ) ,

where a( t ) and b( t ) are binary sequences with symbols (0, 1). Note that b( t ) = s2(t ) and 2a( t ) = s ( t ) - s2(t ) . Using these relationships, one can show after some work (see [l]) that, for Family d with r = 2 s + 1, the binary compo- nents a( t ) and b( t ) are of the form

b ( t ) = t r (we ' ) ,

where w, x E GF(2') and either w or x is nonzero.

It is then straightforward to show that the in-phase and quadrature components, U( t) and U( t), of s( t) are given by

u ( t ) = a ( t ) + b ( t )

Thus, by adding these components, one obtains a binary m-sequence, which could prove important in practice.

Example: Family d is defined as the set of all nonzero solutions of a linear recurrence over Z4 whose characteristic polynomial is a primitive basic irreducible dividing x2'-' - 1 in Z4[x]. Table I lists the primitive basic irreducible polynomials over Z4 up to degree 10 that are suitable for generating Family d. Any of the polynomials of degree r from the table may be used to generate Family d with parameters L = 2' - 1 and M = 2'+ 1. Note that the polynomials of Table I are in 1-1 correspondence with the binary primitive irreducible polynomials.

For example, the polynomial f ( x ) = x 3 + 2 x 2 + x + 3 is the characteristic polynomial of the linear recurrence

~ ( t ) = 2 s ( t - 1) + 3s(t - 2) + ~ ( t - 3 ) , (4.11)

over Z4 and generates Family d with period 7 and size 9. The shift register implementation of (4.11) is diagrammed in Fig. 1. Different sequences in Family d may be generated by loading the shift register with any set of initial conditions not identically zero and cycling the shift register through a full period L. Cyclically distinct members of the family may be found by loading the shift register with triples not previ- ously seen during the generation of prior sequences.

Table II provides a standard enumeration of the cyclically distinct sequences of Family d as generated by the polyno- mial f ( x ) = x 3 + 2 x 2 + x + 3. Family members are iden- tified by their trace representations. Corresponding correla- tion sums are also given in the table. From these, the weight distribution of the family is readily verified: A4 = 2'-' + 2"-' = 3 and N = 2'-' - 2"-' = 1. In the correlation dis- tribution, the peak correlation value 7 occurs 2' + 1 = 9 times, - 1 occurs 2" - 2 = 62 times, - 1 + 2" f 02" oc- curs (2r-2 + 2s-1)(22r - 2) = 186 times each, and - 1 - 2" f 02' occurs (2'-' - 2"-')(2'' - 2) = 62 times each.

V. FAMILY 93 Trace Description: Family 9 is defined as the family

S * ( f ) of linearly recurring sequences over Z4 whose charac- teristic polynomial f ( x ) is a primitive basic irreducible of degree r and order 2(2' - 1) with roots of a special kind in GR(4, r). Before restricting attention to Family 9, we first consider the general case of primitive basic irreducibles and their associated families. Let f ( x ) be an arbitrary primitive basic irreducible in Z4[ x]. As noted in Section IB during the discussion of the Galois automorphism group, f ( x ) has precisely r distinct roots ao, a1; a , arP l satisfying p a i = 8". Since a. is a unit, it may be written as (1 + 26,)P, where 0 is a multiplicative generator of the subgroup G I of

1108 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 38, NO. 3, MAY 1992

TABLE I CHARACTERISTIC POLYNOMIALS FOR LINEAR RECURRENCE DEFINING FAMILY d

Degree 3 Degree 4 Degree 5

Degree 6

Degree 7

Degree 8

Degree 9

Degree 10

1213 10231 100323 130133 100203 1 13201 1 1 10020013 1 1 122323 12122333 13210123 100103 12 1 11 1310321 12 132003 1 132 103001 loooO30203 1020332213 1023112133 11 13303003 1 13233 1203 121 1003133 1231310123 1300013333 1302212123 1321003133 10000203001 100302oooOl 1021333023 1 11113111201 11301031201 11323133321 12102023121 12 13202012 1 12313022111 13002310311 13022212321 13201111111

1323 13201 113013

1110231

10030203 11 131 123 12303213 13212213 100301231 1131201 11 1230131 11

1001011333 1021123003 1 1 10220323 1130312123 1133013203 1211213013 1232100323 1301 110213 1303122003 1322110203 100021021 11 10203103311 10231100111 11120120001 11301210321 113232021 11 12110012311 12 132 120001 1232 1 10323 1 13011013111 1303 120200 1 13203331201

113123

12 1 103 1

10201003 11321 133 1231 1203 132232 13 102231321 121 102121 123 132201

1001233203 1021301 133 1 1 1 1300013 1131003213 1133022333 1213232203 12323 10133 1301 3012 13 1303313333 1323013013 10002123121 10203 122 12 1 10233222121 11120232311 11301320031 11330130201 12 1203 1 132 1 12132203311 1232 122203 1 130 1 123223 1 13033113321 1322321 1031

121003

1301121

10221133 11332133 1233 1333

111002031 121201121 130023 12 1

100223 101 3 1021331123 1111311013 1131003323 12 10032 123 1230 103 133 1232322013 1301 323323 1320322013

100202 1303 1 10211131111 11100113201 1 1 12203 132 1 1 13 1201023 1 11330223121 12 122 130201 12301210311 12331133031 13020010231 13201002031 13230112321

123133

1302001

10233123 11332203 13002003

1 110213 1 1 12 130100 1 1302OO111

1020 1 oooO3 1022121323 1112201133 1131030123 1210220333 12303 13 123 12331 13203 13022 102 13 1320333013

10030023231 102130103 11 1 1 11 11 10231 1 113 101 1031 11321001121 12 10012203 1 12122233201 12311302121 1233313231 1 13022100121 132010213 11 13232003001

Note: For degree 3, the entry 1213 represents the polynomial x3 + 2x2 + x + 3. For definition of the characteristic polynomial, refer to (2.1).

R* and 6, E G , U (0). Applying the Galois automorphism a, shows that

a; = (1 + 2 6 3 P 2 ' ,

for i = 0, l , . . . , r - 1. It is easy to show that ai - aj is a unit whenever i # j . Hence, f ( x ) factors in R [ x ] as

r - 1

r = O f ( x ) = Il ( x - a i ) .

As a notational convenience, let a = (1 + 2 6 ) f i , where 6 = 6ir- ' and a= P2'- ' . Note that, unless 6 = 0, a has multiplicative order 2(2' - 1) in R* and that

In order to emphasize dependence on the root a, the linear recurrence family S ( f ) will also be denoted by Ya. When 6 = 0, Y: is Family d. When 6 # 0, f ( x ) has order 2(2' - 1). The families are new but do not necessarily have correlation properties that are nearly optimal. However, if in addition p{ T(6)) = 1 and p6 # 1, then Ya* is Family 9' and does have nearly optimal correlation properties. As in the case of Family d, the following trace characterization is easily proven.

Theorem 7: For every y E R , the sequence { s,(t))?=, defined by s,(t) A T(ya ' ) is a member of the family x. Conversely, every sequence in Ya is of this form for a unique y E R .

In order to identify family membership, the following notation will be used:

If y is a unit, then s,(t; a) has period equal to the multi- plicative order of a in R*. For nonunits y = 2 v , with v E R * , s,(t; a) = 2tr(F13') is twice a binary m-sequence and has period 2' - 1 regardless of the multiplicative order of a. Thus, when 6 # 0, the family Ye* (from which the zero-divisor sequences are excluded) consists entirely of se- quences of period 2(2' - 1). The number of cyclically dis- tinct sequences in Ya*, for 6 # 0, is

- 2'-1 4'- 2' - 2(2' - 1)

BOZTAS et al.: 4-PHASE SEQUENCES WITH NEAR-OPTIMUM CORRELATION PROPERTIES 1109

Fig. 1. Shift register implementation of family d as generated by charac- teristic polynomial f ( x ) = x3 + 2 x 2 + x + 3.

TABLE II ENUMERATION OF FAMILY &AS GENERATED BY x 3 + 2 x2 + x + 3

Y U t ) !xY)

2 1

- 1 1 - 0 1 - p2

1 - 8 3

1 - p4

1 - 0 5 1 - p6

2002022 3221211 1223233 1013102 1100123 2010333 11 12030 2133003 2303130

- 1 - 3 + 2w - 3 - 2w 1 + 2 w 1 + 2 w 1 - 2 w 1 + 2 w 1 - 2 w 1 - 2w

~ ~

Note: w = exp ( j r 12).

For two sequences s ( t ) and s'(t) of common period L, their interleaving is a sequence of period 2 L denoted here by

s( t)As'( t ) { s(0) , s'(0) , s( 1) ,

s ' ( l ) ; * - , s ( ? ) , s ' ( ? ) , . * - } .

The next proposition shows that the families Ya*, for 6 # 0, consist of special interleavings of sequences from Family d. As a result, the correlations of these families are the sum of two Family d correlation values. Not all families of this type will have near optimum correlation properties. For example, if the two interleaved sequences have the same Family d correlation value, the interleaved sequence will have a correlation sum too large for the family to be nearly optimal. For Family L$?, the interleaved sequences retain nearly optimal correlation properties.

Proposition 2: Let a = (1 + 2 6 ) a with p6 # 0. Then, every sequence in Ya* is an interleaving of two cyclically distinct sequences of Family d. Furthermore, the cyclic equivalence classes of Ya* correspond to a partitioning of the cosets of G , in R* into distinct pairs (yG,, ayG,).

Proof: Let sy( t ; a) with y E R* be a sequence in Ya*. Since a' = 0, it is easy to see that

All cyclic shifts of sy(?; a) are interleavings of either a cyclic shift of sy with a cyclic shift of say or, in the opposite order, a cyclic shift of say with a cyclic shift of sy. Hence, the cyclic equivalence class of s, in Ya* corresponds to the pair (yG,, ayG, ) of cosets of G , in R * . Since a 'yG, = y G I , the pairing is unique and partitions the cosets of G I in R*. Since a is not a power of p, the cosets yG, and ayG, in R* are distinct. 0

Correlation Properties: The remainder of the paper will now focus upon the correlation properties of Family 9. The correlation sums associated with Families d and 97 will be written as

2'- 1

t = 1 ((7; p ) = u T ( Y @ ' ) ,

((7; a) = (5 -2) 2(2'- 1)

t = 1

By (5.1), the correlation sum {(y; a) has value

{(y; a) = ((7; 6) + {(ay; P ) , (5-3)

which is the sum of correlation values associated with cycli- cally distinct sequences from Family d. Since the period of sequences from Family 23 is twice that of Family d, the maximum nontrivial correlation parameter of Family 9 can only be larger by a factor of d? in order for the family to be asymptotically optimal with respect to the Welch bound. Thus, when viewed as vectors, the complex numbers ((7; 0) and {(ay; p) must be nearly orthogonal. This is proven in the next two lemmas.

Lemma 2: For a = (1 + 26) fi with p{ T(6) ) = 1 and p6 # 1, the element 1 - a p t in R may be expressed as

1 - ap t = p=(t+'/2)(1 + 2z ) ,

x ' ( Z ' + 6 ' + l ) + x + Z 2 = O ,

where z E R has projection Z = p z satisfying

in K with x = t9*+'/' and 8 = p6. Consequently, among the cosets (1 - a/3')GI for 0 I t s 2' - 2, t # 2r- ' - 1, the cosets + G I and - aG, each appear once. The remaining cosets appear either twice or not at all depending on whether tr(Z6) = 0 or tr(Z6) = 1, respectively. Furthermore, the coset yG, appears as some (1 - aPt)G1 iff the coset -yG, does not.

Proof: It is easy to show that 1 - ap' is a unit except when T = 2'-' - 1. If T = 2'-' - 1, then o r = l/a and 1 - ap' = -26 # 0 is a zero-divisor. When 1 - aP' is a unit, it may be written as

1 - = p * ( r + l / Z ) (1 + 2z ) , (5.4)

for some Z E R . Recall that the cosets of G I in R* are determined by the term 1 + 2 2 and that 1 + 2 2 and 1 + 22' are in the same coset iff pz = pz'. Squaring both sides of (5.4) yields 1 - 2ap' + p2'+' = P 2 * ( r + 1 / 2 ) , while applying the automorphism U, yields 1 - (1 + 26')p2'+' =

P2r(r+1/2)(1 + 22'). Combining these two results gives the equation 2[6'+'/' - (1 + 6 ' )P2 '+ ' ] = 2z2(1 + /3"+'). By (3. l), it suffices to solve the corresponding equation in K:

x ' ( 2 + s2 + 1) + x + zz = 0,

= 0 7 + 1 / 2 (. # 2r-1 - 1) . ( 5 . 5 )

Note that the side conditions x = Or+'/' and 7 # 2 r - 1 - 1 exclude both x = 0 and x = 1 as legitimate solutions.

1110 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 38, NO. 3, MAY 1992

The cosets not appearing among the (1 - aPr)GI corre- spond to elements 1 + 2 2 for which (5.5) has no solution X E K \ (0, l}. The remaining cosets appear once or twice depending on whether there are one or two distinct solutions of (5.5) in K \ {0,1}. Since 8 # I , the exceptional case x = 1 is never a possible solution. The exceptional case x = 0, however, must be specifically excluded when Z = 0. p u s , when Z = 0, the only solution to (5.5) is x = 1/(1 + S 2 ) . Hence, the corresponding coset + G I appears as 11 - aPT)Gl for precisely one choice of T . When Z = 1 + 6, the quadratic term of (5.5) vanishes. The only solution in this case is x = 1 + 8'. Hence, the corresponding coset -aG, appears only once.

For other values of Z, (5.5) has either two distinct solu- tions in K \ (0, l}(when tr(B8) = 0) or it has none (when tr(Z8) = 1). Note that yG, = (1 + 22)G, implie! --?GI = (1 + 2(2 + l))G,. Hence, when Z#{O, 1 , l + 6, S}, the coset yG, appears as (1 - aPT)G, for precisely two differ- ent choices of 7 iff the coset - y G , never appears for any choice of 7. This follows since tr(6) = 1. Note that tr(Z8) = 1 when Z = 1 or Z = 8, so that, in general, even allowing B E (0, 1, 1 + 8, 8}, the coset y G , appears at least once among the (1 - aPf)G1 iff the coset - y G , never appears.

n

Hence, from (5.7), (5.8), and Lemma 2,

- r(r) - r*(r) - r ( w ) - r * ( 4 , thereby proving (5.6) and establishing the lemma. 0

U

Proposition 4: Let { = ~ : ! ! ; - ' ) o ~ ( y a ' ) with y ER". lies on the circle 12 + t I 2 = 2'+'. Consequently, Lemma 3: k t Y E R * and a = (1 + 26) Ja, with

Pi T ( @ } = 1 and P s f 1. Interpreted as vectors, the com- plex numbers 1 + ((7; 0) and 1 + {(ay; 0) are orthogonal.

Proof: Let r(y) = {(y; p) and {(ay) = {(ay; 0).

Then, the only possible values of { are as follows:

1) if r = 2s, then ( = -2 f 2' f 02'; 2) if r = 2s + 1, then r = -2 f 2'+' or { = -2 f

Note that 1 + {(y) and 1 + f ( a y ) are orthogonal iff w2s+1.

Proof: By (5.3) and Lemma 3,

+ {(ay) + r * ( a y ) + 2 = 0. (5.6) = I l + r ( r ; P ) I ' + I I+ r (ay ;P) I '

Consider the product Then 12 + { 1 ' = 2(2') by Theorem 4. All possible correla- tion values follow by solving a simple Diophantine equation

Paralleling the development of the weight distribution of Family d, we now consider the first and second moments associated with Family g. The first moment sum is

as in Proposition 1. I7

4

C { ( T i ; a) = C I(ri; P ) + C r(aYi; P ) m* ?,,er* yi=r *

(5.9) = g=(x ) + J ( X ) = 0 . ER* XER*

t = O Using (4.8), the second moment sum may be written

2'-2 2'-2

r=O f=O + J ( Y ( I - u B r ) B ' )

7+21-1 - 1

BOZTAS et 01. : 4-PHASE SEQUENCES WITH NEAR-OPTIMUM CORRELATION PROPERTIES

r(r; a) = ,

As in Lemma 2 , one can show that 1 + is a unit except when 7 = 2'-' - 1. Hence, for fixed 7 # 2'-' - 1, sum- ming the bracketed term on the right over all coset represen- tatives y i € r gives zero by (3.5). For the first term on the right, note that 1 + aP"-'-' = 2(1 + 6) # 0 since p6 # 1. Then, by ( 3 3 , the first term is

w=(Yi(l+aS*'-l-l)x) J ( x ) = - 1. XEG, XER\R*

X # O

Hence, the second moment sum has value

I

2(2' - l ) , - 2 ,

- 2 + 2' + 02' ,

for y = 0, for 2' - 1 nonunits

for ( 2 ' ~ ~ + 2'-')

for ( 2 ' ~ ~ + 2'-l)

for (2'-2 - 2'- l )

for ( 2 ' ~ ~ - 2'-l)

Y Z O ,

- ( 2 ' - 1) units y ,

~ ( 2 ' - 1) units y,

. (2 ' - 1) units y,

-(2' - 1) units y.

- 2 + 2' - w2',

- 2 - 2' + w2',

-2 - 2' - w2',

\

(5.10)

r ( y ; a ) = '

, 2(2' - l ) , for y = 0, - 2 , for2' - 1

- 2 + 2'+', nonunits y # 0,

for (2r-2 + 2')

for (2'-2 - 2')

for 2'-2(2' - 1) units y,

for ~ ' - ~ ( 2 ' - 1) units y.

.(2' - 1) units y,

- 2 - 2'+',

.(2' - 1) units y,

- 2 + w2'+l, - 2 - w2'+l,

\

Pro08 For case l), one lets

For case 2) , one lets

M = 1 { y d * : r(y; a) = - 2 + 2'+l} 1 ,

N = ) { y ~ r * : r ( y ; a ) = - 2 - 2 ' + ' } 1 ,

P = I {yer*: c ( ~ ; a) = -2 + w2s+1} 1

= I { ? E r * : r(y; a) = - 2 - w2s+1} 1 .

1111

9

From the first moment sum (5.9), the second moment sum (5.10), and the fact that I I'*I = 2', the following sets of simultaneous equations may be derived as in the proof of Theorem 5. When r = 2s ,

2 ( M + N ) = 2',

2 ( M + N ) = 2 ' ( M - N ) .

When r = 2 s + 1,

M + N + 2 P = 2',

M + N + 2 P = 2 ' ( M - N ) ,

M + N - 2 P = 0.

Solving these equations gives the claimed weight distribu- tions. U

Let {sA: AEA} be a complete set of cyclically distinct sequences from Family 9? with A = {A,, &, &; * * , &,-I}

C R*. Then, by Proposition 2, A U a A is a complete set of distinct coset representatives for the cosets of GI in R*. As in the proof of Theorem 6, the correlation distribution of Family 9? follows directly from the weight distribution once the number of solutions to the equation

AiaT - Ai = y,

X i , X i € A ,

0 I 7 < 2(2' - 1)

is known for each y E R .

(5.11)

1112 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 38, NO. 3, MAY 1992

I

2(2' - l ) ,

- 2 , - 2 + 2" + w2",

- 2 + 2" - w 2 " ,

- 2 - 2" + w 2 " ,

- 2 - 2" - w2",

,

Theorem 9 (Correlation Distribution): For Family 9,

1) If r = 2s, then

the correlation distribution is as follows.

I

2(2' - l ) ,

- 2 , - 2 + 2 s + 1 ,

2) If r = 2 s + 1 , then

- 2 + w2"+1,

- 2 - w2"+,, I

2'-1 times,

times, 22'-1 - 2'-1

(2'-2 + 2"-1)

(2'-2 + 2"-1)

(2'-2 - 2 s - 7

( 2 - 2 - 2 s - 7

.(22'-' - 2 ') times,

. (227-1 - 2 ') times,

. (22-1 - 2 ') times,

.(2"-' - 2') times.

times, 2'-1

22'-1 - 2'-1 times, (2'-2 + 2")

2") ( 2 ' 4 - .(2"-' - 2') times,

. (22'- 1 - 2 ') times,

2r-2(22r-1 - 2') times,

2r-2 (22r -1 - 2') times.

Pro08 Fix Aj E A. Then, for each y E R , if there is a solution to (5.11), it is unique. Thus, the number of triples ( i , j , 7 ) satisfying (5.11) for fixed Aj is equal to the number of y E R for which solutions exists. Note that (5.11) has a solution iff y # - Aj + 2 v for any v E R . Since the - Aj + 2v are all units, every nonunit y admits a unique solution. Hence, for each Aj E A , the correlation value 2(2' - 1) corresponding to y = 0 appears once in the correlation distri- bution. The correlation sum - 2 appears 2' - 1 times, once for each nonzero ER \ R*. Then, as Aj varies over A, there are a total of 2'-' triples having correlation value 2(2' - 1) and 2'-'(2' - 1) triples having correlation value - 2.

For the units in R , note that - Ai + 2 v and - Xi + 2 v' are in different cosets of G , in R* whenever pv # pv'. Thus, for fixed Aj E A, each coset of G , in R* has 2' - 2 choices of y that yield valid solutions to (5.11). Therefore, as Aj varies over A, each coset of G, gives rise to 2'-'(2' - 2) triples with its associated correlation value. The corre- lation distribution then follows in this case by multiplying

0 Example: Family !ZI is defined as the family Y *(f) of

linearly recurring sequences over Z4 whose characteristic polynomial f ( x ) is a primitive basic irreducible having root CY = (1 + 26)& with p(T(6 ) ) = 1 and pL8 # 1 . Table 111

M , N , and P of Theorem 8 by 2"-I - 2'.

I'

U s(t-1) s(t-2) s(t-3)

Fig. 2. Shift register implementation of family 9 as generated by charac- teristic polynomial f( x ) = x 3 + x + 1.

TABLE IIf SELECTED CHARACTERISTIC POLYNOMIALS FOR FAMILY 9

Degree 3 Degree 4 Degree 5 Degree 6 Degree 7 Degree 8 Degree 9 Degree 10

101 1 12213 100101 120201 3 1220 100 1 122231103 1 m 1 0 2 2 1 10020201003

~ ~~

Note: Entry 1011 represents the polynomial x 3 + x + 1. For definition of the characteristic polynomial, refer to (2.1).

TABLE IV ENUMERATION OF FAMILY 9 AS GENERATED BY x 3 + x + 1

Y sJt ; ff) ( (7; ff)

1 302121 11221013 - 2 + 4 w - 1 10232333223031 - 2 - 4 0 1 - P 13001033120123 2 1 - 8 2 13100301132032 2

Note: w = exp (jr 12) and P 3 + 2P2 + + 3 = 0.

provides representative polynomials of this type for degrees 3 to 10. (Of course, the list is by no means exhaustive. Neither have we tried to identify the characteristic polynomial of fewest nonzero coefficients.) The polynomial of degree r in the table may be used to generate Family 37 with parameters

For example, the polynomial f ( x ) = x3 + x + 1 is the L = 2(2' - 1) and M = 2'-'.

characteristic polynomial of the linear recurrence

s ( t ) = 3s( t - 2) + 3s(t - 3 ) , (5.12)

over Z4 and generates Family !ZI with period 14 and size 4. The shift register implemyntation of (5.12) is diagrammed in Fig. 2. Different sequences in Family !ZI may be generated by loading the shift register with different initial conditions and cycling through a full period L. Initial conditions con- sisting only of zero-divisors in Z4, however, are prohibited.

Table IV provides an enumeration of the cyclically distinct sequences of Family 53 as generated by the polynomial f( x ) = x3 + x + 1. Family members are identified by their trace representations. Corresponding correlation sums are also given in the table. Recalling that each equivalence class of Family !ZI corresponds to a unique pair of cosets of G, in R*, the weight distribution of the family is readily verified from the table: M = 2'-' + 2" = 4, N = 2'-' - 2" = 0, and P = 2'-' = 2. In the correlation distribution, the peak correlation value 14 occurs 2'-' = 4 times, - 2 occurs

- 2'-' = 28 times, - 2 + 2'+' occurs (2'-' + 22'- 1

BOZTAS et al. : 4-PHASE SEQUENCES WITH NEAR-OPTIMUM CORRELATION PROPERTIES 1113

29(2*‘-’ - 2‘) = 96 times, and -2 f w2’+’ occurs (2r -2 ) (22r -1 - 2‘) = 48 times each.

ACKNOWLEDGMENT The authors would like to thank A. Tietiivainen for draw-

ing attention to the paper by P. Sol&

[31

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