1582 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 42, NO. 5, SEFTEMBER 1996

Improved Binary Codes and Sequence Families from 24-Linear Codes

Abhijit G. Shanbhag, P. Vijay Kumar, Member, IEEE, and Tor Helleseth, Member, IEEE

Abstract-A recently derived bound on exponential sums over Galois rings is used to construct a nested chain of 24-linear binary codes and binary sequences. When compared with the chain of Delsarte-Goethals’ codes, the codes in the new chain offer a larger minimum distance for the same code size. The binary sequence families constructed also make use of Nechaev’s construction of a cyclic version of the Kerdock code. For a given value of maximum correlation, the binary sequences are shown to have family size considerably larger than the best sequence families known.

Index Terms- Kerdock codes, Delsarte-Goethals’ codes, CDMA se- quences, exponential sums.

I. INTRODUCTION The Delsarte-Goethals codes DG (m, T ) [3] , m odd, m 2 3 , 1 5

r 5 m - 112, are a nested chain of nonlinear binary codes which may be regarded as generalizations of the Kerdock code. In [4], Hammons et al. showed that the Kerdock code, the Delsarte-Goethals codes, and certain other efficient nonlinear codes such as the “Preparata” and “Goethals” codes could be described simply as the images under the Gray map of &-linear codes. This mapping also provided an explanation for the apparent duality of Kerdock and Preparata codes. Further, the Kerdock and Delsarte-Goethals codes have a simple trace description when viewed from the 24 domain.

Earlier, in [8], Nechaev showed that the Kerdock code punctured in two coordinates could be made cyclic through a coordinate permutation. The permutation was identified by viewing the Kerdock code as the code derived from a Z4-linear code via the most significant bit representative.

The present correspondence uses a bound for exponential sums over Galois rings of characteristic 4, derived by Kumar et al. [6] and Helleseth et al. [5] , to construct a new nested chain of nonlinear binary codes and sequences. The chain of codes are the Gray-mapped images of a new chain of 24-linear codes. The codes DG (m, T ) for T = 1 , 2 are also contained in the new code chain. Corresponding to each code DG (m, T ) , 3 5 T 5 m - 112, we show that there is a code in the new chain, of the same length and size as DG (m, T ) , but whose minimum distance is significantly larger.

By shortening the codes in the new chain of binary codes and then applying Nechaev’s permutation, one obtains equivalent and efficient nonlinear cyclic codes. This construction is relevant to a research problem in MacWilliams and Sloane [7, Research Problem 2.61 which asks for examples of cyclic nonlinear codes. The first member in this cyclic code chain was first constructed by Nechaev.

Manuscript received July 14, 1994; revised April 3, 1996. This work was supported in part by the National Science Foundation under Grant NCR-93- OS017 and the Norwegian Research Council under Grants 107542/410 and 107623/420.

A. G. Shanhhag and P. V. Kumar are with the Communication Sciences Institute, Electrical Engineering-Systems, University of Southern California, Los Angeles, CA 90089-2565 USA.

T. Helleseth is with the Department of Informatics, University of Bergen, H@yteknologisenteret, N-5020 Bergen, Norway.

Publisher Item Identifier S 0018-9448(96)054S2-1.

These cyclic codes lead to a chain of efficient and nested sequence families. The first three members in the chain improve upon the small and large set of Kasami sequences and the modified Gold family constructed by Rothaus [9]. The first member in the chain of sequence families was independently constructed by Udaya and Siddiqi [ 101 who were unaware of the results of Nechaev. The remaining members in the sequence chain are new.

After an initial version of this paper was prepared, we came to know that Barg [l], using the results in [4] and [8], gave another description of the first two members in the sequence chain.

Section II provides the background on Galois rings needed to discuss Zg-linear codes. The new chain of nonlinear binary codes is constructed and compared with the Delsarte-Goethals chain in Sec- tion 111. The construction of the nonlinear binary sequence families and comparisons with existing sequence families appears in Section IV.

11. PRELIMINARIES

A. Galois Rings of Characteristic 4

Let m 2 1 be a fixed integer. The Galois ring R = GR (4, m) of size 4” is the unique Galois extension of degree m over 24. R is a local ring having unique maximal ideal M = 2R. Let p:R + RIM denote the (mod2) reduction map given by

p(z) = z + M , z E R.

Thus p ( R ) = R I M E F where F is the finite field with 2” elements.

As a multiplicative group, the units R* in R have the following group structure:

R’ Z Zzm-1 x Zz x Zz x . . . x 2 2 . P

m copies

Let ,3 be a primitive element of R*, by which we will mean that B is a gen_erator of the cyclic group Zzm-1. Define 7 = {0,1,,5’,. . . , p2 -’}. It can be shown that every element z E R has a unique 2-adic expansion

z = a + 2b, a , b E 7. The Galois group of RI24 is cyclic of order m generated by the

automorphism o given by

o(z) = a2 + 2b2

where x = a + 2b E R and a , b E 1. The trace mapping Tr (.): R + 24 is then defined via

Tr(z) = z + a ( z ) + . . . + ( ~ ” - ~ ( z ) .

B. Exponential Sums over Galois Rings

that f(.) is not expressible in the form Let f ( z ) t R[z] be a nondegenerate polynomial. By this we mean

f (z) = a ( s ( z ) ) - s(z) + 8

for any g(z) E R,[T],% E R,. For any such

f(.) = fdZ Z=O

we introduce the following degree-related parameters:

D I J = max(il0 5 i 5 n , f z E R\2R}

Dz , f = max(il0 5 i 5 n, f t E 2R, fi # 0) .

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The weighted degree D f of the polynomial f (x) is defined via

l?f = max{2D1,~ , D z , ~ } . (1) Theorem 1 in [6] then states:

Theorem I :

I Remark: It was asmmed in [6] that fi = 0, when i is even, in the

above theorem. However, the theorem can be shown to hold for the more general case priesented above.

It is evident from Theorem 1 that

where X(z) denotes the real part of x. However, after the initial preparation of this paper, the authors (with 0. Moreno) [SI showed that this bound can be strengthened up to a factor of fi in some specific instances as shown below.

For any positive integer j , let w2(j) denote the Hamming weight of the binary expansion of j. For

11.

Then

We define

Thus

Example: Let

f (x) = yx + e, c E R, y E R', m odd.

Here D f = wz(f(x)) = 2, e f = (m + 1)/2, h f = (m - 1)/2. It follows from Theorem 2 that the obvious bound

is improved to

Given an integer D 2 1, we define

SO = { f €1 R[z]JDf 5 D , f t 10 fo r i even}.

Clearly, each polynomial in SO is nondegenerate. Then we have:

Lemma 1: Let D 2 1 be an integer. Then lsDl = 2(D-LD/4l)m

where 1x1 denotes the largest integer 5 2. Proofi It is easy to show that

/ { U E T [ Z ] I U E SO)/ = 2 4 1 ~ ~ 1 - 1 ~ / 4 ~ )

I{b E 7 [ ~ ] 1 2 b E S,}l =2m(D-LD'2').

(3) (4)

Thus since each f E R[z] can be expressed uniquely as f = a + 2b, a, b E 7[z], we have from (3) and (4)

lsDl ~ 2(D-t0/4l)m. 0

111. CODES BETTER THAN THE DELSARTI-GOETHALS CODES A linear ( n , M ) quaternary code C4 is a submodule of Z," of

size M . Let a = ( a o , . .. , a n - l ) , b = (bo , . . . , b n - l ) be any two codevectors in C4. We define an inner product on C4 via

p ( a , b ) = X (;Io1 Waz--b, ), w = Gi.

The Lee weights of 0 ,1 ,2 ,3 E Z4 are 0,1,2,1, respectively. The Lee norm of any vector in 2: is the sum of the Lee weights of the components. We note that the Lee weight w ~ ( y ) of an element y E 2 4 can be expressed as

wI,(y) = 1 - R(w").

Thus the Lee distance d L ( a , b ) = W L ( U - b ) , a ,b E C4, is related to p ( a , b ) via

(5) &(a, 6) = n - p(a, b) . Next, consider the (nonlinear) map 4: 2 4 --$ Z,"

0 i o 0

1 4 0 1

2 4 1 1

3 + 10.

This map, called the Gray map, can be viewed as the composite of two binary maps T , U so that

d(z) = ( 4 x ) , u ( 4 ) , 2 E 2 4 .

4: z,n -2;"

Extending 4 in a natural way, we obtain a map

c + ( 4 c h 4.)). Clearly, (under the Hamming metric).

inner product between distinct codewords in C4, i.e.,

pmax = max{p(a, b)la, b E C4, a # b } .

is an isometry from Z t (under the Lee metric) to 2,".

Given a linear quaternary code C4, let p,,, denote the maximum

As a result of (5) and the isometry of the Gray map, we have: Theorem 3: Let Cq be a &-linear code and C = 4(C4) be its

image under the Gray map. Then the minimum distance d,,, of the binary code C is given by

d,,, = n - max{p(a,b) la ,b E C4,a # b }

= n - pmax.

The Delsarte-Goethals codes DG (m, T ) [ 3 ] are a nested chain of nonlinear binary codes which may be regarded as generalizations of the Kerdock code. It was shown in [4] that the Delsarte-Goethals codes are the images under the Gray map of a chain DG4 (m, T ) of 24 -linear codes. The following theorem provides the details.

1584 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 42, NO. 5 , SEPTEMBER 1996

r

3 4 5 6

D, Common dmin 2 dmin Polynomial description Code Size (C(m, D,)) (DG(m, r ) ) of C(m, Or)

6 25mf2 2" - (5)2? 2m - (4&)2? E + AZ + B Z ~ + 2 c x 5 7 26mf2 2m - (6)2? 2m - (842)2? E + A x + B x 3 + 2Cxs + 2Ex7 9 27mf2 2" - ( 8 ) 2 f 2" - (16&)2? E + A x + B x 3 + 2Cx5 + 2Ex7 + 2Fx9 10 28mf2 2m - (9)2? 2" - ( 3 2 d 2 ) 2 7 E + A x + B x 3 + C x 5 + 2Ex7 + 2Fx9

Theorem4: Let m 2 3 be odd and 1 5 T 5 m - 1/2. Let DG4 ( m , r ) denote the quatemary code

DG4 (m, T ) = { ~ x , ~ l A E R x I,, t E 24)

where the (A, c)th codeword C X , ~ = (c~ ,< (z ) , ic E I) has the trace description

c ~ , ~ ( z ) = Tr(A0z) + 2 Tr ( X , Z ~ + ~ ' ) + E,

Z = l

XO E R, A, E I, 1 5 i 5 r.

The rth element D G ( m , r ) , in the chain of binary, nonlinear Del- sarte-Goethals codes is then given by

DG (m, r ) = d(DG4 (m, r ) ) and has parameters

length = 2"+', size = 4m+12rm, d,,, = 2" - 2'-1/2 . fi. Using Theorem 2, we now construct a different nested chain of

nonlinear codes which are the Gray-map images of a new chain of &-linear codes.

Given an integer D , 1 5 D 5 2/2m, let C4(m,D) denote the 24 -linear code

C4(m,D) = {(Tr ( f ( z ) )+E ,z E 1 ) 1 ~ E & , f ( Z ) E SD} . Note that the condition D 5 @ forces the exponents of the monomial terms with nonzero coefficents in f ( ~ ) to lie in distinct 2-cyclotomic cosets. Further let e g = min{eflf E Sn} and h D = min{hflf E S D } . Let

Clearly

n(m, D ) 5 ( D - 1)G.

From Lemma 1, we have that C4(m, D ) is of size 4 . 2(D-LD/4i)m. The inner product of any two codewords (Tr (fl(z)),ic E I), (Tr(fi(z)) ,z E 7 ) is bounded, using Theorem 2, via

I 4 m , D )

Pmax 5 & ( m , D ) (6)

since the weighted degree of fl - f2 5 D . Thus

and using Theorem 3, the minimum distance d,,, of C ( m , D ) = q5(C4(m, D ) ) has the lower bound

d,,, 2 2" - n(m, D).

The results are summarized below.

1 5 D I 2 m / 2 . Then the binary codes Theorem 5 Let m 2 3 be odd. Let D be an integer satisfying

C(m, D ) = d(C4(m, D ) )

of length 2"+l have parameters

dmin 2 2" - K ( ~ , D ) /cym, D ) I = 4 . 2(D-LD/4J)m.

It follows from Theorems 4 and 5 that for given D , the codes DG (m, T ) and C(m, D ) have the same size whenever

r = D - - 2

or equivalently, choosing the smallest value of D when there are two values of D for a given r

We then have Corollary I : Let m 2 3 be odd. For a given integer r , 1 5 r 5

m - 1/2, let the integer D, be given by D , = [r + 1/31 + r + 2. Then the binary codes D G ( m , r ) and C(m,D,) have the same size 4"+'2'", but different minimum distances dmin given by 2" - 2r--(1'2)2mL/2 and 2 2" - n(m, Dr), respectively.

In Table I, a polynomial description of C(m, Dr) is given. For example, since the polynomial description of C(m,6) is f ( z ) = E + Az + Bz3 + 2Cx5, the set of codewords in C(m, 6) is given via

C(m,6) = {{T~(E+AZ+BZ~+~CZ~)IA,B,E E R,C E 7}5t~}. Table I compares the minimum distance of the codes DG (m, r )

and C(m, O r ) having the same size for r = 3,4,5, and 6. Since computation of ~ ( m , D,) depends upon the actual value of m and D,, we have used the upper bound ~ ( m , D T ) 5 ( D , - 1)@ in the table. For values of D, = 2,3,5, the codes C(m, D r ) coincide with the Kerdock code and the first two members of the DG chain, respectively. However, we obtain a considerably higher (lower bound for) minimum distance for the succeeding members of C(m, D T ) as compared to DG (m, r ) having the same size.

Example: Let m = 7 , r = 3 (or D, = 6) . From Theorem 4, the code DG (7 ,3) has parameters

length = 256, size = 237, amin = 64.

Corresponding to the code C(7,6), we have h7 = l , e7 = 2 . Thus ~ ( 7 , 6 ) = 54. It follows from Theorem 5, that the code C(7,6) has parameters

length = 256, size = 237 , dmi, 2 74.

I v . NONLINEAR CYCLIC BINARY CODES AND

SEQUENCES VIA NECHAEV'S PERMUTATION

A. ESJicient Nonlinear Cyclic Binary Codes

code of length 2" - 1 given by Given an integer D 2 1, let Cz(m, D ) be the &-linear, cyclic

C4*(m, D ) = {es l f 6 SDI where

cf = (Tr [f(z)], z = P t , t = 0,1,2, . . . ,2" - 2).

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 42, NO. 5 , SEPTEMBER 1996 1585

Thus C $ ( m , D ) may he regarded as being obtained by shortening C4(m,D) with respect to the coordinate x = 0.

From Theorem 5 it follows that the binary code C * ( m , D ) = d(C2 ( m , D ) ) obtained by projecting the 24-linear code via the Gray map, has parameters

length = 2"+l - 2 , size = 2(0-L0'4')m, d,,, 2 2" - n(m, D ) . We now extend Nechaev's [SI analysis of the Kerdock code and show that the code C * ( m , D) is equivalent via a permutation of coordinates, to a cyclic code.

Consider the binary code

E'(m,D) = k f l f E S D I with

where the map 7 r ( . ) was defined earlier in context with the Gray map. Let

Tr[f(Pt)]=a(t)S2b(t) , a ( t ) , b ( t ) E { O , l } , 0<t12m-2 be the base-2 expansion of Tr [ f ( P t ) ] . Since

where 0 5 t 5 2"+l - 3 and denotes (mod2) addition, it follows that the cyclic code P ( m , D ) is equivalent to C * ( m , D ) under the permutation map q acling on the set {0,1,. . . ,2n - l}, n = 2" - 1, given by

(1,n + 1)(3,n + 3) . . . (n - 2 , 2 n - 2 )

i.e., q(1) = n + 1, etc. Define for some u ( t ) , q ( a ( t ) ) = a ( q ( t ) ) . Then this definition implies that, for Tr [ f ( P t ) ] E C*(m, D )

(7 0 d)(Tr[f(Pt) l ) = ~ ( 3 ~ Tr[f(Pt))l) E P ( m , D ) . The permutation map q will be referred to henceforth as Nechaev's permutation. Research Problem 2.6 in MacWilliams and Sloane [7] asks for constructions of efficient nonlinear cyclic codes. An application of Theorcm 5 and Nechaev's permutation gives new examples, details are given in Theorems 6 and 7.

Theorem 6: Let P( m , D ) be the binary code given by

l ' (m,D) = {sflf E S D I

where

sf=((17045)(Tr[f(Pt)]) , t=0,1,...,2"+' - 3 )

and q denotes Nechaev's permutation. Then P ( m , D ) is nonlinear in general, and cyclic with parameters

length = 2"'l - 2, size = 2(DpLD/4')m, d,,, 2 2" - n(m, D ) . Theorem 6 will be used in the next subsection on efficent binary sequences.

Repeating the above steps, beginning with the code C(m, D ) punctured in the coordinate x = 0 rather than shortened, we then obtain

Theorem 7: Let N ( m , D ) be the binary cyclic code given by

N ( m . D ) = {nf+lf E SD,E E 24) where

1 ~ f , ~ = ( (TI 0 (rr [.T(P')] + E ) , t = o , ~ . . . ,2"+* - 3 ) .

Then N ( m , D ) is nonlinear in general, with parameters

length=2"+'-2, size=2(D-LD/41)m+2, d,,, 2 2"-n(m, D)-2.

The code N(m, 2) was previously constructed by Nechaev [SI and shown to be equivalent to the punctured Kerdock code.

B. Binary Sequences with Low Correlation

We next extract from the cyclic code P ( m , D ) a large family of binary sequences of period 2(2" - 1) having desirable correlation properties. In the following theorem on periodicity of sequences, we denote the period of any sequence y ( t ) by Nv.

Theorem 8: Let s ( t ) be a 24 sequence with period N , , N , odd, such that p ( s ( t ) ) # 0. Define

u( t ) = 3ts( t ) , t E 2 and

p ( t ) = T ( U ( t ) ) , t E 2.

Then u(t) and p ( t ) have period 2 . N, . Proofi Since the sequence { 3 t } t E z (mod4) has period 2

Nu = 1.c.m. ( 2 , N S ) = 2NS.

Clearly, Np I 2NS. Now let

s ( t ) = a ( t ) + 2 b ( t ) , a ( t ) , b ( t ) E (0, l} vt E 2

b ( t ) = n(s ( t ) )

and let c ( t ) = a ( t ) CE b ( t ) . Thus

c ( t ) = 7r(3S(t)).

We consider the following two cases: 1) N p is even. Then

7r(3ts(t)) = 7r(31fNPS(t + N p ) )

= 7r(3ts(t + N p ) ) . It then follows that

b ( t ) = b ( t + N p ) c( t ) = c( t + N p ) . Thus NsINp and so N p = 2Ns since N , is odd.

2) N, is odd. Here

7r(3tS(t)) = 7r(3t+"s(t + N p ) )

= 7r(3t+ls(t + N,) ) . We then have

b ( t ) = c(t + N p ) c( t ) = b ( t + Np). Thus Ns12Np which implies N,IN, and so b ( t ) = c( t ) . This implies p ( s ( t ) ) = 0 which contradicts the hypothesis.

Thus N p = 2Ns. 0 Remark: When p ( s ( t ) ) = 0, Nu = N, . As a consequence of Theorem 8, we have the following corollary

which identifies a necessary and sufficent condition under which the sequence corresponding to a codeword in P ( m , D ) will have maximal period 2(2- - 1).

Corollary 2: Let

f(Z) = E R[z]\2R[Z], f E SO z

for some integer D 2 1. Set

p ( t ) = 7r(3tTr[f(Pt)]) , W E 2.

Then p ( t ) has period 2(2" - 1) iff

g.c.d. { { ~ l f ~ # O}, Z m - 1) = 1.

We now associate a sequence with each codeword in P(m, D) in the usual manner and denote the resulting family of sequences by P ( m , D ) again. Two sequences s z ( t ) , sJ ( t ) in P(m, D ) are said to be cyclically distinct if there is no integer r,O 5 r 5 2m+1 - 3 for which

s z ( t + 7 ) = S J ( t ) , vt, 0 5 t 5 2"+l - 3

1586

Family Period L

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 42, NO. 5, SEPTEMBER 1996

Size of Maximum Correlation Family Value

Kasami (Small Set)

S(m, 2)

I I I

2" - 1, m even 2 7 x Lfr 2 7 + 1 M L:

2(2" - l), m odd 2m-lx 4 VJ5 .2?+2= L:

Kasami (Large Set)

S(m, 3)

I I I

I I I 2m - 1, m even 2?(2m + 1) N L%

2'"-' N $

2 . 2 t + 1 N 2 . Lfr

2 4 5 - 2 7 + 2 M 2 . L? 2(2" - l), m odd

Modified Gold (Rothaus)

S(m,5)

2" - 1, m even M L: 4 . 2 ? + 1 x 4L:

2(2" - l ) ,m odd 23m-1 = % 4 & - 2 ? + 2 ~ 4 L :

where t + T is computed modulo 2(2" - 1). Let the set

S(m, D ) c P(m, D ) (8) consist of a maximal family of pairwise, cyclically distinct sequences in P(m, D ) with each sequence having period 2(2" - 1). The following theorem gives a precise expression and a useful lower bound for the size of S ( m , D ) . We let p denote the Mobius function given by

i f d = 1 if d is the product of T distinct primes p(d ) = (-l)r, { :: otherwise.

Theorem 9: Given integers m 2 3 and D , 2 5 D 5 2"/', let S ( m , D ) be as defined above. The size IS(m, D)I of this family satisfies

(9) ~ ) l 2 2(D-lD/41--l)m--l

Further

- 2( L ( " / d ) l - L ( D / 2 d ) ! ) - ) p ( d ) .

(10) Proof: Consider the following subset of P(m, D ) :

where

S f ( t ) = (7 O 4) (TrIf(P71) = (Tr [fi . ( W t l + 3t Tr [g(Pt)) l )

where g(z) = f (z) - flz. Let SLd(m, D ) denote the maximal set of pairwise cyclically distinct sequences in S'(m, D ) . The sequences

in this set have period 2(2" - 1) by virtue of Corollary 2. Thus

s:d(m, D ) c S(m, D ) . Using Lemma 1, and noting that 3p has order 2(2" - l), we have

- - 2(D-LD/41 -1)"-1

Thus I S ( ~ , D ) I 2 2(D-lD/41-l)m-l,

To obtain an exact expression for the size lS(m, D)l in (lo), we note from Corollary 2 that p ( t ) has period 2(2" - 1) iff p ( f ( z ) ) # 0 and the greatest common divisor of the exponents in f(x) with nonzero coefficents is relatively prime to 2" - 1. Further, the condition that f (z) be nondegenerate and have weighted degree 5 2m/2, causes the exponents in f(z) with nonzero coefficents to lie in distinct 2-cyclotomic cosets modulo 2" - 1. It follows then that

where p is the Mobius function. 0 E The correlation C2,3 ( T ) between the sequences s z ( t ) , s3 ( t )

S(m, D ) at shift r, 0 5 r 5 2"+l - 3 is

1=0

The maximum nontrivial correlation Cmax(m, D ) of S ( m , D ) is then defined via

Cmax(m, D ) = m={lCz,,(?-)IIst(t),

s3( t ) E S(m, D ) , either i # 3 or T # O}.

With notation as above, we prove

1587 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 42, NO. 5, SEPTEMBER 1996

Theorem IO: The maximum nontrivial correlation parameter

Cmax(m,D) =2.max{p(a,b)la,b E C,*(m,D)> (11)

(12) Proof: Let s l ( t ) , s j ( t ) be any two sequences in S ( m , D ) . Let

&,%.(m, D ) of S(m, D ) satisfies

5 2 . rc(m, D ) + 2.

a:, a3 E C,*(m, D ) be the unique codewords satisfying

(7) 0 $) (a i ) = ( S % ( t + T ) , 0 5 t 5 2“+l - 3)

(7 ) 0 $ ) ( U j ) = ( S j ( t ) , O 5 t 5 2”+l - 3).

It then follows, from definition of ( T )

< t , j ( T ) =2(2” - 1) - - 2 2 0 H ( S t ( t $ T ) - Sj(t)10 5 t 5 2”” - 3 )

= 2(2” - 1) -- 2 ’ W L ( U : . - a,)

= 2 . p(a:.,a,).

Thus

Smax(m, D ) *I 2 . n ia{p(a , b)la, b E c;(m, D ) ) . The result now follows from (6). 0

Table I1 compares the family size and the correlation properties of S ( m , D ) with some other sequences. Note that S(m,2), S(m,3) , and S(m,5) are the images under the composition of the Gray map and Nechaev’s permutation of Family A [2], DG: (m, 1) and DG; (m, 2), respectively. From Theorem 10, we have

Cmmax(mr D ) 5 2n(m. D ) + 2 = 2 ( D - 1)2”-”’ + 2, m odd, D = 2,3,5.

From Table I1 we note that S(m, 2), S(m, 3), and S(m, 5) have the same maximum correlation (normalized by length) but a larger family size by a factor on the order of l /z as compared to the Small and Large sets of Kasami sequences and the modified Gold sequences [9], respectively.

Exumple: Let m = 9, D = 2. From Theorem 9, S(9,2) has 256 cyclically distinct sequences each of period 1022. From Theorem 10, the maximum nontrivial correlation of S (m,2 ) is at most 34. The comparable Small set of Kasami sequences has period 1023 and the maximum nontrivial correlation is 33. The family size, however, is only 32.

It is straightforwardl to construct Family S(m, 2) from Nechaev’s construction of a cyclic code equivalent to the punctured Kerdock code. Udaya and Siddiqi, who were unaware of the results of Nechaev, present an independent construction of Family S(m, 2) in u01.

REFERENCES

A. Barg, “On small families of sequences with low periodic correlation,” in Lecture Notes in Computer Science, vol. 781. Berlin, Germany: Springer-Verlag, 1994, pp. 154-158. S. Boqtas, A. R. Hammons, Jr., and P. V. Kumar, “4-phase sequences with near-optimum correlation properties,” IEEE Trans. Inform. Theory, vol. 38, no. 3, pp. 1101-1113, May 1992. P. Delsarte and J. M. Goethals, “Altemating bilinear forms over GF (4):’ J. Comb. Theory, Ser. A, vol. 19, pp. 26-50, 1975. A. R. Hammons, P’. V. Kumar, A. R. Calderbank, N. J. A. Sloane, and P. Sole, “The 24-lmearity of Kerdock, Preparata, Goethals, and related codes,” IEEE Tran,s. Inform. Theory, vol. 40, no. 2, pp. 301-319, Mar. 1994. T. Helleseth, P. V. Kumar, 0. Moreno, and A. G. Shanbhag, “Improved estimates via exponential sums for the minimum distance of Z4-linear trace codes,” IEEE Trans. Inform. Theory, vol. 42, no. 4, pp. 1212-1216, July 1996.

[6] P. V. Kumar, T. Hellcscth, and A. R. Calderbank, “An upper bound for Weil exponential sums over Galois rings and applications,” IEEE Trans. Inform. Theory, vol. 41, no. 2, pp. 456468, Mar. 1995.

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vol. I, pp. 365-384, 1991.

On the Weight Hierarchy of Kerdock Codes Over 2,

Kyeongcheol Yang, Member, IEEE, Tor Helleseth, Member, IEEE, P. Vijay Kumar, Member, IEEE, and Abhijit G. Shanbhag

Abstract-The 7th generalized Hamming weight d, of the Kerdock code of length 2” over Zq is considered in this correspondence. A lower bound on d, is derived for any T, and d, is exactly determined for T = 0.5,1,1.5,2,2.5. In the case of length 22m, d, is determined for any T , where 0 5 T 5 m and 2r is an integer. In addition, it is shown that it is sometimes possible to determine the generalized Hamming weights of the Kerdock codes of larger length using the results of d, for a given length. This correspondence also provides a closed-form expression for the Lee weight of a Kerdock codeword in terms of the coefficients in its trace expansion.

Index Terms-Linear codes over 24, Kerdock codes, weight hierarchy, generalized Hamming weights.

I. INTRODUCTION Let 2 4 be the ring of integers modulo 4. A linear code over 2 4 with

blocklength n is an additive subgroup of 2,“. The Lee weights of the elements 0 , 1 , 2 , 3 of Z4 are 0 ,1 ,2,1, respectively. The Lee weight of a vector a E 2 4 is defined to be the sum of the Lee weights of its components. Hammons, Kumar, Calderbank, Sloane, and Sol6 [3] have shown that efficient nonlinear codes such as Kerdock, Preparata, etc., can be very simply constructed as binary images under the Gray map of linear codes over 2 4 .

The Galois ring R = GR (4, m) is an extension of 2 4 of degree m. R is a local ring having a unique maximal ideal M = 2R and the quotient ring RIM is isomorphic to FP where F2m is a finite field with 2” elements (see [3], [4] for details).

Manuscript received November 3, 1995; revised March 22, 1996. The material in this correspondence was presented in part at the Generalized Hamming Weight Workshop, Germany, February 1996. This work was supported in part by the Korean Ministry of Information and Communications, the Norwegian Research Council under Grants 107542/410 and 107623/4120, and the National Science Foundation under Grant NCR-93-05017.

K. Yang is with the Department of Electronic Communication Engineering, Hanyang University, Seoul 133-791, Korea.

T. Helleseth is with the Department of Informatics, University of Bergcn, Hgytcknologisenteret, N-5020 Bergen, Norway

P. V. Kumar and A. G. Shanbhag arc with the Communication Sciences Institute, Electrical Engineering-Systems, University of Southern California, Los Angeles, CA 90089-2565 USA.

Publisher Item Identifier S 0018-9448(96)05604-0.

0018-9448/96$05.00 0 1996 IEEE