Construction of a (64, 2 37, 12) Code via Galois Rings

Transcription

1 Designs, Codes and Cryptography, 10, (1997) c 1997 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands. Construction of a (64, 2 37, 12) Code via Galois Rings A. R. CALDERBANK AT&T Labs-Research, Murray Hill, New Jersey GARY MCGUIRE* Department of Mathematics, University of Virginia, Charlottesville, VA Communicated by: D. Jungnickel Received January 24, 1995; Revised April 2, 1996; Accepted June 25, 1996 Dedicated to Hanfried Lenz on the occasion of his 80 th birthday. Abstract. Certain nonlinear binary codes contain more codewords than any comparable linear code presently known. These include the Kerdock and Preparata codes, which exist for all lengths 4 m 16. At length 16 they coincide to give the Nordstrom-Robinson code. This paper constructs a nonlinear (64, 2 37, 12) code as the binary image, under the Gray map, of an extended cyclic code defined over the integers modulo 4 using Galois rings. The Nordstrom-Robinson code is defined in this same way, and like the Nordstrom-Robinson code, the new code is better than any linear code that is presently known. Keywords: Algebraic coding theory, codes over rings 1. Introduction The Kerdock, Delsarte-Goethals, Goethals and Preparata codes are nonlinear binary codes that contain more codewords than any comparable linear code presently known. For example, the Preparata code contains twice as many codewords as the extended 2-error-correcting BCH code of the same length, and the Goethals code contains 4 times as many codewords as the extended 3-error-correcting BCH code of the same length. Recently Hammons et al. [5] showed that these codes can be very simply constructed as binary images under a certain natural map, called the Gray map, of linear codes over Z 4 (although this requires a slight modification of the Preparata and Goethals codes). We refer the reader to [5] for information concerning Galois rings and their use in the construction of cyclic codes over Z 4. We refer the reader to [6] and [8] for the original definitions of the Kerdock and Preparata codes. Section 2 concerns the doubly transitive group of affine permutations of the finite field F 2 m, given by x px + q, where p, q F 2 m and p 0. This group leaves invariant the extended e-error-correcting BCH code of length 2 m and its dual. We use Galois rings to define a class of extended linear cyclic codes over Z 4 that are also left invariant by this group. One family of such codes is obtained from the dual of the extended 2-errorcorrecting BCH code by Hensel lifting. We explain how to use the action of the affine group to simplify calculation of Lee weight enumerators in the Z 4 domain. For m odd, * Research supported in part by AT&T Research.

2 158 CALDERBANK AND MCGUIRE this requires a description of the orbits of the affine group on codewords in the dual of the extended binary 2-error-correcting BCH code of length 2 m, and this information is presented in Theorem 2.3. In Section 3 we introduce a family of extended cyclic codes of length 2 m (where m is odd) that contain 2 5m+2 codewords. The Gray image of the Z 4 -dual is a (2 m+1, 2 2m+1 5m 2 ) code and here we look for good error correcting properties for certain values of m. When m = 5, we show that this Gray image is a (64, 2 37, 12) code, which is better than the [64, 37, 10] code listed by Brouwer and Verhoeff [1]. This is an exceptional code; for m 5 the Gray image is a (2 m+1, 2 2m+1 5m 2, 8) code (see Calderbank et al. [3] for details). 2. The Action of the Affine Group Let θ be a primitive element of F 2 m, and let m i (x) denote the minimal polynomial of θ i over the binary field F 2. Let C2 e denote the e-error correcting BCH code of length 2m 1 generated by (the lcm) m 1 (x)...m 2e+1 (x). Thus C2 1 is the Hamming code of length 2m 1. Let Ĉ2 e denote the extended code of length 2m obtained from C2 e by adding an overall parity check. The dual code (Ĉ2 1) is then the first order Reed-Muller code RM(1,m) of length 2 m. We use the field F 2 m to index the 2 m coordinate positions in the usual way, with the zero element indexing the position of the overall parity check. It is well known (see MacWilliams and Sloane [7], Chapter 8) that the extended codes Ĉ2 e are invariant under the doubly transitive group G of affine permutations of F 2 m given by x a x + b, (2.1) where a, b F 2 m and a 0. Let T ={0,1,ξ,ξ 2,...,ξ 2m 2 }, where ξ is a primitive (2 m 1)th root of unity in the Galois ring GR(4 m ). Following Hammons et al. [5] we can describe G in terms of T rather than the field F 2 m.nowgconsists of permutations of T given by x (ax + b) 2m, (2.2) where a, b T and a 0. Note that µ(ax + b) = µ((ax + b) 2m ) = ax + b, so that (2.1) and (2.2) are describing the same permutation. The order of G is 2 m (2 m 1). The next theorem is a generalization of Theorem 20 of Hammons et al. [5]. THEOREM 2.1 Let be a system of linear equations over Z 4 in the variables c x, that consists of c x = 0, (2.3) c x x = 0, (2.4)

3 CONSTRUCTION OF A (64, 2 37, 12) CODE VIA GALOIS RINGS 159 together with sets of 3 equations given by ( ) 2 c x x 2 j 1 +1 = 0, (2.5) c x x 2 j +1 = 0, (2.6) ( ) 2 c x x 2 j+1 +1 = 0, (2.7) where j runs through any set of positive integers. Then the set of solutions (c x ) to the linear system is invariant under the affine group G. Proof. Repeated application of the Frobenius automorphism to (2.4) gives c x x 2 j = 0, (2.8) for all j. Now c x ((ax + b) 2m 2j +1 ) = c x (ax + b + 2a 2m 1 x 2m 1 b 2m 1 ) 2j +1 = c x (ax + b) 2j (ax + b + 2a 2m 1 x 2m 1 b 2m 1 ) = c x (a 2j x 2j + b 2j + 2a 2j 1 x 2j 1 b 2j 1 ) (ax + b + 2a 2m 1 x 2m 1 b 2m 1 ) = c x (a 2j +1 x 2j +1 + a 2j x 2j b + 2a 2m 1 +2 j x 2m 1 +2 j b 2m 1 + axb 2j + b 2j a 2m 1 x 2m 1 b 2m 1 +2 j + 2a 2j 1 +1 x 2j 1 +1 b 2j 1 + 2a 2j 1 x 2j 1 b 2j 1 +1 ). Equations (2.3) through (2.8), together with the observation ( ) 2 2 x 2m 1 +2 j = 2 x 2 j+1 +1 = 0, imply that each of the 8 individual sums appearing above is equal to zero. Similar arguments show that each of the sums c x (ax + b) 2m, 2 c x ((ax + b) 2m ) 2j 1 +1, and 2 c x ((ax + b) 2m ) 2j+1 +1 are equal to zero, and details can be found in Theorem 20 of Hammons et al. [5].

4 160 CALDERBANK AND MCGUIRE COROLLARY 2.2 The linear code D 4 over Z 4 with parity check matrix ξ ξ 2 ξ i ξ 2m ξ 3 ξ 6 ξ 3i ξ 3(2m 2) 0 2 2ξ 5 2ξ 10 2ξ 5i 2ξ 5(2m 2) is invariant under the affine group G. Proof. Apply Theorem 2.1 with j = 1. The Z 4 -linear Kerdock code K and Preparata code P constructed by Hammons et al. [5] are } K = {(T (λx) + ɛ) λ GR(4 m ), ɛ Z 4 and its dual code P = K. Here T : GR(4 m ) Z 4 is the trace map. Thus, P is the Z 4 -linear code consisting of the vectors (c x ), c x Z 4 such that c x = 0 and c x x = 0. The Gray image φ(k) is the standard Kerdock code [6] but the Gray image φ(p) differs from the standard Preparata code [8], in that it is not a subcode of the extended Hamming code, but of a nonlinear code with the same weight distribution as the extended Hamming code. Remark. Note that the linear code over Z 4 with parity check matrix ξ ξ 2 ξ i ξ 2m ξ 3 ξ 6 ξ 3i ξ 3(2m 2) need not be invariant under the affine group G. If m is odd, then the weight distribution of the dual of the extended binary 2-errorcorrecting BCH code Ĉ 2 2 of length 2m is given by i A i m 1 2 (m 1)/2 2 m 1 (2 m 1) 2 m 1 2 m (2 m 1) + 2(2 m 1) = (2 m + 2)(2 m 1) 2 m (m 1)/2 2 m 1 (2 m 1) 2 m 1 (2.9)

5 CONSTRUCTION OF A (64, 2 37, 12) CODE VIA GALOIS RINGS 161 Every codeword in the code D 4 defined in Corollary 2.2 is congruent modulo 2 to a codeword in (Ĉ 2 2 ). The orbit structure of G on the codewords of (Ĉ 2 2 ) will be used to simplify the calculation of the complete weight enumerator of D 4. THEOREM 2.3 Let m be odd, let (Ĉ2 2) be the dual of the extended 2-error-correcting binary BCH code of length 2 m, and let G be the affine group of order 2 m (2 m 1). (i) The group G partitions the codewords of weight 2 m 1 into two orbits. One orbit contains the 2(2 m 1) codewords of weight 2 m 1 in the first order Reed-Muller code (Ĉ2 1). The other orbit has size 2 m (2 m 1), and G is sharply transitive on this orbit. (ii) The group G acts transitively on the codewords of minimum weight 2 m 1 2 (m 1)/2, and it acts transitively on the codewords of weight 2 m (m 1)/2. Proof. (i) The subcode of (C2 2) generated by the reciprocal of (x 2m 1 1)/m 1 (x) is a simplex code, and adding an overall parity check produces a codeword c in (Ĉ2 2) of the form c = (0, tr(λ), tr(λθ),...,tr(λθ 2m 2 )), for some λ F 2 m. If the affine permutation x ax + b of F 2 m fixes the codeword c, then tr(λ(a + 1)x) = tr(λb) is constant, so that a = 1 and b λ ={y F 2 m tr(λy) = 0}. Note that if b λ, then the translation x x + b will map c to its complement 1 + c. There are 2(2 m 1) codewords in the orbit of G that contains c, and these are the codewords of weight 2 m 1 in the first order Reed-Muller code (Ĉ2 1). The subcode of (C2 2) generated by the reciprocal of (x 2m 1 1)/m 3 (x) is a second simplex code, and adding an overall parity check produces a codeword c in (Ĉ2 2) of the form c = (0, tr(λ), tr(λθ 3 ),...,tr(λθ 3(2m 2) )), for some λ F 2 m. If the affine permutation x ax + b of F 2 m fixes c, then tr(λ(a 3 + 1)x 3 ) = tr(λ(a 2 bx 2 + axb 2 + b 3 )), for all x F 2 m. The map x tr(λ(a 3 + 1)x 3 ) is a quadratic form on F 2 m, and the map x tr(λ(a 2 bx 2 + axb 2 + b 3 )) is affine, so m odd and equality forces a = 1 and tr((λb + λ 2 b 4 )x 2 ) = tr(λb 3 ), for all x F 2 m. Now λb + λ 2 b 4 = 0, and if b 0, then λb 3 = 1 and tr(λb 3 ) = 0, which contradicts the assumption that m is odd. Hence the action of the affine group G produces 2 m (2 m 1) distinct images of c. We have now accounted for all the remaining codewords of weight 2 m 1, so the proof of part (i) is complete. (ii) Every codeword in (Ĉ 2 2 ) is of the form c ɛ,α,β = (ɛ + tr(αx) + tr(βx 3 )) x F2 m,

6 162 CALDERBANK AND MCGUIRE where ɛ = 0 or 1, and α, β F 2 m. We shall now derive conditions on α, β that imply c ɛ,α,β is fixed by a unique translation x x + d. This codeword c ɛ,α,β will then have weight 2 m 1 ± 2 (m 1)/2, since part (i) shows that the stabilizer of a codeword of weight 2 m 1 in the affine group G has order 1 or 2 m 1. From part (i) we may suppose β 0. If c ɛ,α,β is fixed by the translation x x + d, then tr(x 2 (βd + β 2 d 4 )) = tr(αd + βd 3 ), for all x F 2 m, so that tr(αd + βd 3 ) = 0 and βd 3 = 1. Note that since m is odd, the equation βd 3 = 1 always has a unique solution d = β 1/3. The restriction that m is odd also implies that tr(αd) = tr(αβ 1/3 ) = tr(βd 3 ) = tr(1) = 1. Note that if c ɛ,α,β is fixed by a second affine permutation x ax + b, then it is also fixed by the translation x x + da 1. We may conclude that if c ɛ,α,β is fixed by a unique translation, then the stabilizer of c ɛ,α,β in G has order 2. The number of codewords c ɛ,α,β with a stabilizer of order 2 in G is 2 2 m 1 (2 m 1), since there are 2 choices for ɛ,2 m 1 choices for β 0, and given β, there are 2 m 1 choices for α that will give tr(αβ 1/3 ) = 1. These codewords fall into 2 orbits as described in part (ii). Note that there are 2 2 m 1 (2 m 1) codewords c ɛ,α,β for which tr(αβ 1/3 ) = 0. These are the codewords of weight 2 m 1 that form the large orbit described in part (i). 3. The Calculation of Weight Enumerators In this section, we take m = 5, and we calculate weight enumerators of the code D 4 of length 32 over Z 4. The complete weight enumerator (or c.w.e.) of C is cwe C (w, x, y, z) = a C w n 0(a) x n 1(a) y n 2(a) z n 3(a), where n i (a) is the number of components of a that are congruent to i modulo 4. Since a monomial transformation may change the sign of a component, the appropriate weight enumerator for an equivalence class of codes is the symmetrized weight enumerator (or s.w.e.) given by swe C (w, x, y) = cwe C (w, x, y, x). The MacWilliams identity over Z 4 expresses the symmetrized weight enumerator of the dual code C in terms of swe C (w, x, y): swe C (w, x, y) = 1 C swe C(w + 2x + y,w y,w 2x +y). The binary code (Ĉ 3 2 ) is a [32, 16, 8] code which we shall denote by B. The subcode of D 4 consisting of codewords with all entries either 0 or 2 is 2B. Every codeword d D 4

7 CONSTRUCTION OF A (64, 2 37, 12) CODE VIA GALOIS RINGS 163 Table 1. Enumeration of codewords in D4 modulo 2. with a given symmetrized Lee composition according to congruence

8 164 CALDERBANK AND MCGUIRE Table 2. Enumeration of codewords in D 4 with a given symmetrized Lee composition.

9 CONSTRUCTION OF A (64, 2 37, 12) CODE VIA GALOIS RINGS 165 is congruent modulo 2 to a binary codeword d in (Ĉ2 2). The set of all codewords in D4 that are congruent to d modulo 2 is given by d + 2B d where B d is a coset of B that depends on d. The affine group permutes the subsets d + 2B d, and this action preserves the Hamming weight wt H (d). By Theorem 2.3 there are 6 orbits, and the complete Lee weight enumerator of D4 can be calculated as the weighted sum of complete Lee weight enumerators of representative subsets d + 2B d. The six orbits correspond to d having Hamming weight 0, 12, 16, 20, 32 where there are two orbits on codewords d with Hamming weight 16, and one orbit on all others. When d has Hamming weight 0 or 32 the Lee weight enumerator of the coset is determined by the Hamming weight enumerator of B. The Lee weight enumerators of the remaining cosets were calculated by direct enumeration after finding an appropriate coset representative d. The complete Lee weight enumerator of D4 is denoted by cwe D (w, x, y, z), and by 4 Theorem 2.3 it is given by cwe D (w, x, y, z) = cwe 2B(w, x, y, z) + cwe 4 2B (z,w,x,y) +62cwe 16 (w, x, y, z) cwe 16 (w, x, y, z) cwe 12 (w, x, y, z) cwe 12 (z,w,x,y). Setting z = x, we obtain the symmetrized Lee weight enumerator listed in Table 1. We apply the MacWilliams identity over Z 4 to obtain the symmetrized Lee weight enumerator of D 4 listed in Table 2. Remarks. The Gray image of D4 is a (64, 227, 16) code, which is no better than the [64, 27, 16] code that appears in Brouwer and Verhoeff [1]. However the Gray image of D 4 is a (64, 2 37, 12) code, which is better than the [64, 37, 10] code that appears in Brouwer and Verhoeff [1]. This code looks even better when compared with the extended 5-error-correcting BCH code, which is a [64, 33, 12] code. More detailed information on the cosets of D4, including graphs of the correlation values, can be found in the technical memorandum [2]. References 1. A. E. Brouwer and T. Verhoeff, An updated table of minimum distance bounds for binary linear codes, IEEE Trans. Inform. Theory, Vol. 39 (1993) pp A. R. Calderbank and Gary McGuire, Construction of a (64, 2 37, 12) code via Galois Rings, AT&T Bell Laboratories Technical Memorandum BL (1994). 3. A. R. Calderbank, G. McGuire, P. V. Kumar, and T. Helleseth, Cyclic codes over Z 4, locator polynomials and Newtons identities, IEEE Trans. Inform. Theory, Vol. 42 (1996) pp A. R. Calderbank and N. J. A. Sloane, Modular and p-adic cyclic codes, Designs, Codes and Cryptography, Vol. 6, (1995) pp A. R. Hammons Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé, The Z 4 -linearity of Kerdock, Preparata, Goethals and related codes, IEEE Trans. Inform. Theory, Vol. 40 (1994) pp A. M. Kerdock, A class of low-rate nonlinear binary codes, Inform. and Control, Vol. 20 (1972) pp F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam (1977). 8. F. P. Preparata, A class of optimum nonlinear double-error correcting codes, Inform. and Control, Vol. 13 (1968) pp