A 2-adic approach to the analysis of cyclic codes

This paper describes how 2-adic numbers can be used to analyze the structure of binary cyclic codes and of cyclic codes defined over Z _2(a), a⩾2, the ring of integers modulo 2^a. It provides a 2-adic proof of a theorem of McEliece that characterizes the possible Hamming weights that can appear in a binary cyclic code. A generalization of this theorem is derived that applies to cyclic codes over Z_2(a) that are obtained from binary cyclic codes by a sequence of Hensel lifts. This generalization characterizes the number of times a residue modulo 2^a appears as a component of an arbitrary codeword in the cyclic code. The limit of the sequence of Hensel lifts is a universal code defined over the 2-adic integers. This code was first introduced by Calderbank and Sloane (1995), and is the main subject of this paper. Binary cyclic codes and cyclic codes over Z_2(a) are obtained from these universal codes by reduction modulo some power of 2. A special case of particular interest is cyclic codes over Z₄ that are obtained from binary cyclic codes by means of a single Hensel lift. The binary images of such codes under the Gray isometry include the Kerdock, Preparata, and Delsart-Goethals codes. These are nonlinear binary codes that contain more codewords than any linear code presently known. Fundamental understanding of the composition of codewords in cyclic codes over Z₄ is central to the search for more families of optimal codes. This paper also constructs even unimodular lattices from the Hensel lift of extended binary cyclic codes that are self-dual with all Hamming weights divisible by 4. The Leech lattice arises in this way as do extremal lattices in dimensions 32 through 48     

On algebraic decoding of the Z4-linearCalderbank-McGuire code

The quaternary Calderbank-McGuire (see Des., Codes Cryptogr., vol.10, no.2, 1997) code is a Z₄-linear code of length 32 which has 2³⁷ codewords and a minimum Lee distance of 12. The Gray map of this code is known to be a nonlinear binary (64, 2³⁷,12) code. The Z₄-linear Calderbank-McGuire code can correct all errors with Lee weight ⩽5. An algebraic decoding algorithm for the code is presented in this paper. Furthermore, we discuss an alternative decoding method which takes advantage of the efficient BCH decoding algorithm     

Cyclic codes and self-dual codes over F2+uF2

We introduce linear cyclic codes over the ring F₂+uF ₂={0,1,u,u¯=u+1}, where u²=0 and study them by analogy with the Z₄ case. We give the structure of these codes on this new alphabet. Self-dual codes of odd length exist as in the case of Z₄-codes. Unlike the Z₄ case, here free codes are not interesting. Some nonfree codes give rise to optimal binary linear codes and extremal self-dual codes through a linear Gray map     

On algebraic decoding of the Z4-linear Goethals-likecodes

The Z₄-linear Goethals-like code of length 2^m has 2^2m+1-3m-2 codewords and minimum Lee distance 8 for any odd integer m⩾3. We present an algebraic decoding algorithm for all Z₄-linear Goethals-like codes C_k introduced by Helleseth et al.(1995, 1996). We use Dickson polynomials and their properties to solve the syndrome equations     

Two 16-state, rate R=2/4 trellis codes whose free distances meetthe Heller bound

For rate R=1/2 convolutional codes with 16 states there exists a gap between Heller's (1968) upper bound on the free distance and its optimal value. This article reports on the construction of 16-state, binary, rate R=2/4 nonlinear trellis and convolutional codes having d _free=8; a free distance that meets the Heller upper bound. The nonlinear trellis code is constructed from a 16-state, rate R=1/2 convolutional code over Z₄ using the Gray map to obtain a binary code. Both convolutional codes are obtained by computer search. Systematic feedback encoders for both codes are potential candidates for use in combination with iterative decoding. Regarded as modulation codes for 4-PSK, these codes have free squared Euclidean distance d_{E, free}²=16     

On coset weight distributions of the Z4-linear Goethalscodes

We study the coset weight distributions of two well-known families of codes: the three-error-correcting binary Z₄-linear Goethals codes of length N=2^m+1, m⩾3 odd, and the Z₄ -linear Goethals codes over Z₄ of length n=N/2=2^m . The hard case is the weight distributions of cosets of weight 4. To know the weight distribution of the coset of weight 4 we have to know the number of codewords of weight 4 in such a coset. Altogether, there are nine different types of cosets of weight 4. For six cases, we give the exact expressions for the number of codewords of weight 4, and for three other cases, we give such expressions in terms of Kloosterman sums     

Improved estimates via exponential sums for the minimum distance ofZ4-linear trace codes

An upper hound for Weil-type exponential sums over Galois rings was derived by Kumar, Helleseth, and Calderbank (see ibid., vol.41, no.3, p.456, 1995). This bound leads directly to an estimate for the minimum distance of Z₄-linear trace codes. An improved minimum-distance estimate is presented. First, McEliece's result on the divisibility of the weights of binary cyclic codes is extended to Z₄ trace codes. The divisibility result is then combined with the techniques of Serre (1983) and of Moreno and Moreno (see ibid., vol.40, no.11, p.1101, 1994) to derive the improved minimum-distance estimate. The improved estimate is tight for the Kerdock code as well as for the Delsarte-Goethals codes     

The Z4-linearity of Kerdock, Preparata, Goethals, andrelated codes

Certain notorious nonlinear binary codes contain more codewords than any known linear code. These include the codes constructed by Nordstrom-Robinson (1967), Kerdock (1972), Preparata (1968), Goethals (1974), and Delsarte-Goethals (1975). It is shown here that all these codes can be very simply constructed as binary images under the Gray map of linear codes over Z₄, the integers mod 4 (although this requires a slight modification of the Preparata and Goethals codes). The construction implies that all these binary codes are distance invariant. Duality in the Z₄ domain implies that the binary images have dual weight distributions. The Kerdock and “Preparata” codes are duals over Z₄-and the Nordstrom-Robinson code is self-dual-which explains why their weight distributions are dual to each other. The Kerdock and “Preparata” codes are Z₄-analogues of first-order Reed-Muller and extended Hamming codes, respectively. All these codes are extended cyclic codes over Z₄, which greatly simplifies encoding and decoding. An algebraic hard-decision decoding algorithm is given for the “Preparata” code and a Hadamard-transform soft-decision decoding algorithm for the I(Kerdock code. Binary first- and second-order Reed-Muller codes are also linear over Z₄ , but extended Hamming codes of length n⩾32 and the Golay code are not. Using Z₄-linearity, a new family of distance regular graphs are constructed on the cosets of the “Preparata” code     

On the weight hierarchy of Preparata codes over Z4

Hammons et al. (see ibid., vol.40, p.301-19, 1994) showed that, when properly defined, the binary nonlinear Preparata code can be considered as the Gray map of a linear code over Z₄, the so called Preparata code over Z₄. We consider the rth generalized Hamming weight d_r(m) of the Preparata code of length 2^m over Z₄. For any m⩾3, d_r(m) is exactly determined for r=0.5, 1, 1.5, 2, 2.5 and 3.0. For a composite m, we give an upper bound on d_r(m) using the lifting technique. For m=3, 4, 5, 6 and 8, the weight hierarchy is completely determined. In the case of m=7, the weight hierarchy is completely determined except for d₄(7)     

Quasi-cyclic codes over Z4 and some new binarycodes

Previously, (linear) codes over Z₄ and quasi-cyclic (QC) codes (over fields) have been shown to yield useful results in coding theory. Combining these two ideas we study Z ₄-QC codes and obtain new binary codes using the usual Gray map. Among the new codes, the lift of the famous Golay code to Z₄ produces a new binary code, a (92, 2²⁴, 28)-code, which is the best among all binary codes (linear or nonlinear). Moreover, we characterize cyclic codes corresponding to free modules in terms of their generator polynomials     

Quaternary Convolutional Codes From Linear Block Codes Over Galois Rings

From a linear block code B over the Galois ring GR(4, m) with a k times n generator matrix and minimum Hamming distance d, a rate-k/n convolutional code over the ring Z₄ with squared Euclidean free distance at least 2d and a nonrecursive encoder with memory at most m - 1 is constructed. When the generator matrix of B is systematic, the convolutional encoder is systematic, basic, noncatastrophic and minimal. Long codes constructed in this manner are shown to satisfy a Gilbert-Varshnmov bound.     

An Upper Bound on the Minimum Distance of Serially Concatenated Convolutional Codes

This correspondence presents an upper bound on the minimum distance of serially concatenated codes with interleaver where the inner code is a systematic recursive convolutional encoder and the outer code is any convolutional encoder. The resulting expression shows that the minimum distance of the concatenated code cannot grow more than O(K¹-1/d_f ^(O)), where K is the information word length, and d_f ^(O) is the free distance of the outer code. The obtained upper bound is shown to agree with and, in some cases, improve over previously known results     

Studying the locator polynomials of minimum weight codewords of BCHcodes

Primitive binary cyclic codes of length n=2^m are considered. A BCH code with designed distance δ is denoted B(n,δ). A BCH code is always a narrow-sense BCH code. A codeword is identified with its locator polynomial, whose coefficients are the symmetric functions of the locators. The definition of the code by its zeros-set involves some properties for the power sums of the locators. Moreover, the symmetric functions and the power sums of the locators are related to Newton's identities. An algebraic point of view is presented in order to prove or disprove the existence of words of a given weight in a code. The principal result is the true minimum distance of some BCH codes of length 255 and 511. which were not known. The minimum weight codewords of the codes B(n2^h -1) are studied. It is proved that the set of the minimum weight codewords of the BCH code B(n,2^m-2-1) equals the set of the minimum weight codewords of the punctured Reed-Muller code of length n and order 2, for any m     

Bounds on the Minimum Homogeneous Distance of the pr-ary Image of Linear Block Codes Over the Galois Ring GR(pr,m)

In this correspondence, bounds are derived on the minimum homogeneous distance of the image of a linear block code over the Galois ring GR(p^r,m), with respect to any basis of GR(p^r,m). These bounds depend on the parameters of GR(p^r ,m), the minimum Hamming distance of the block code, and the average value of the homogeneous weight applied on the base ring Zopf _p ^r. Examples are given of Galois ring codes that meet these bounds     

Combinatorial bounds for list decoding

Informally, an error-correcting code has "nice" list-decodability properties if every Hamming ball of "large" radius has a "small" number of codewords in it. We report linear codes with nontrivial list-decodability: i.e., codes of large rate that are nicely list-decodable, and codes of large distance that are not nicely list-decodable. Specifically, on the positive side, we show that there exist codes of rate R and block length n that have at most c codewords in every Hamming ball of radius H^-1(1-R-1/c)·n. This answers the main open question from the work of Elias (1957). This result also has consequences for the construction of concatenated codes of good rate that are list decodable from a large fraction of errors, improving previous results of Guruswami and Sudan (see IEEE Trans. Inform. Theory, vol.45, p.1757-67, Sept. 1999, and Proc. 32nd ACM Symp. Theory of Computing (STOC), Portland, OR, p. 181-190, May 2000) in this vein. Specifically, for every ε > 0, we present a polynomial time constructible asymptotically good family of binary codes of rate Ω(ε⁴) that can be list-decoded in polynomial time from up to a fraction (1/2-ε) of errors, using lists of size O(ε^-2). On the negative side, we show that for every δ and c, there exists τ < δ, c₁ > 0, and an infinite family of linear codes {C_i}_i such that if n_i denotes the block length of C_i, then C _i has minimum distance at least δ · n_i and contains more than c₁ · n_i^c codewords in some Hamming ball of radius τ · n_i. While this result is still far from known bounds on the list-decodability of linear codes, it is the first to bound the "radius for list-decodability by a polynomial-sized list" away from the minimum distance of the code     

Negacyclic codes of length 2/sup s/ over galois rings

Codes over the ring of integers modulo 4 have been studied by many researchers. Negacyclic codes such that the length n of the code is odd have been characterized over the alphabet Zopf₄, and furthermore, have been generalized to the case of the alphabet being a finite commutative chain ring. In this paper, we investigate negacyclic codes of length 2^s over Galois rings. The structure of negacyclic codes of length 2^s over the Galois rings GR(2^a,m), as well as that of their duals, are completely obtained. The Hamming distances of negacyclic codes over GR(2^a,m) in general, and over Zopf₂ ^a in particular are studied. Among other more general results, the Hamming distances of all negacyclic codes over Zopf₂ ^a of length 4,8, and 16 are given. The weight distributions of such negacyclic codes are also discussed     

Algebraic construction of cyclic codes over Z8 with agood Euclidean minimum distance

Let S(8) denote the set of the eight admissible signals of an 8PSK communication system. The alphabet S(8) is endowed with the structure of Z₈, the set of integers taken modulo 8, and codes are defined to be Z₈-submodules of Z₈ⁿ. Three cyclic codes over Z₈ are then constructed. Their length is equal to 6, 8, and 7, and they, respectively, contain 64, 64, and 512 codewords. The square of their Euclidean minimum distance is equal to 8, 16-4√2 and 10-2√2, respectively. The size of the codes of length 6 and 7 can be doubled while the Euclidean minimum distance remains the same     

An analysis of Chen's construction of minimum-distance five codes

In 1991, C.L. Chen used the inverted construction Y₁ on binary linear codes of minimum Hamming distance five to construct a new [47, 36, 5] code. We examine this construction in depth and show that no such codes are obtained unless the fields GF(8) or GF(32) are used. Using MAGMA, we prove that the binary [11, 4, 5] code and the binary [15, 7, 5] extension found by Chen are the only possible such codes using the field GF(8); indeed, the latter is a Bose-Chaudhuri-Hocquenghem (BCH) code. We prove also that, using the field GF(32), precisely three nonequivalent binary [47, 36, 5] codes arise along with one extension to a [63, 51, 5] code     

Optimal binary linear codes and Z4-linearity

We give necessary and sufficient conditions for a binary linear code to be Z₄-linear. Especially we treat optimal, binary linear codes and determine all such codes with minimum weight less or equal to six which are Z₄-linear     

有限域上最优码的一种新的构造方法

基于域F_p^m上一类特殊的矩阵,定义了环R(_p^m,k)=F_p^m[u]/k>到F_p^p_mj的一个新的Gray映射,其中u^k=0、p为素数、j为正整数且p^j－1+1≤k≤p^j.得到了环R(_p^m,k)上码长为任意长度N的(1+u)常循环码的Gray象是F_p^m上长为p^jN的保距线性循环码,并给出了Gray象的生成多项式,构造了F₃,F₅和F₇上的一些最优线性循环码.