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)     

On the second generalized Hamming weight of the dual code of adouble-error-correcting binary BCH code

The generalized Hamming weight of a linear code is a new notion of higher dimensional Hamming weights. Let C be an [n,k] linear code and D be a subcode. The support of D is the cardinality of the set of not-always-zero bit positions of D. The rth generalized Hamming weight of C, denoted by d_r(C), is defined as the minimum support of an r-dimensional subcode of C. It was shown by Wei (1991) that the generalized Hamming weight hierarchy of a linear code completely characterizes the performance of the code on the type II wire-tap channel defined by Ozarow and Wyner (1984). In the present paper the second generalized Hamming weight of the dual code of a double-error-correcting BCH code is derived and the authors prove that except for m=4, the second generalized Hamming weight of [2^m-1, 2m]-dual BCH codes achieves the Griesmer bound     

A New Upper Bound on the Block Error Probability After Decoding Over the Erasure Channel

Motivated by cryptographic applications, we derive a new upper bound on the block error probability after decoding over the erasure channel. The bound works for all linear codes and is in terms of the generalized Hamming weights. It turns out to be quite useful for Reed-Muller codes for which all the generalized Hamming weights are known whereas the full weight distribution is only partially known. For these codes, the error probability is related to the cryptographic notion of algebraic immunity. We use our bound to show that the algebraic immunity of a random balanced m-variable Boolean function is of order ^m/₂(1-o(1)) with probability tending to 1 as m goes to infinity     

Cyclic codes over Z4, locator polynomials, and Newton'sidentities

Certain nonlinear binary codes contain more codewords than any comparable linear code presently known. These include the Kerdock (1972) and Preparata (1968) codes that can be very simply constructed as binary images, under the Gray map, of linear codes over Z₄ that are defined by means of parity checks involving Galois rings. This paper describes how Fourier transforms on Galois rings and elementary symmetric functions can be used to derive lower bounds on the minimum distance of such codes. These methods and techniques from algebraic geometry are applied to find the exact minimum distance of a family of Z ₄. Linear codes with length 2^m (m, odd) and size 2(2^m+1-5m-2). The Gray image of the code of length 32 is the best (64, 2³⁷) code that is presently known. This paper also determines the exact minimum Lee distance of the linear codes over Z₄ that are obtained from the extended binary two- and three-error-correcting BCH codes by Hensel lifting. The Gray image of the Hensel lift of the three-error-correcting BCH code of length 32 is the best (64, 2³²) code that is presently known. This code also determines an extremal 32-dimensional even unimodular lattice     

Translates of linear codes over Z4

We give a method to compute the complete weight distribution of translates of linear codes over Z₄. The method follows known ideas that have already been used successfully by others for Hamming weight distributions. For the particular case of quaternary Preparata codes, we obtain that the number of distinct complete weights for the dual Preparata codes and the number of distinct complete coset weight enumerators for the Preparata codes are both equal to ten, independent of the code length     

On Z4-duality

Recently the notion on binary codes called Z₄-linearity was introduced. This notion explains why Kerdock codes and Delsarte-Goethals codes admit formal duals in spite of their nonlinearity. The “Z₄-duals” of these codes (called “Preparata” and “Goethals” codes) are new nonlinear codes which admit simpler decoding algorithms than the previously known formal duals (the generalized Preparata and Goethals codes). We prove, by using the notion of exact weight enumerator, that the relationship between any Z₄-linear code and its Z₄ -dual is stronger than the standard formal duality and we deduce the weight enumerators of related generalized codes     

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     

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 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     

Bounds on the minimum support weights

The minimum support weight, d_r(C), of a linear code C over GF(q) is the minimal size of the support of an r-dimensional subcode of C. A number of bounds on d_r(C) are derived, generalizing the Plotkin bound and the Griesmer bound, as well as giving two new existential bounds. As the main result, it is shown that there exist codes of any given rate R whose ratio d_r/d₁ is lower bounded by a number ranging from (q^r-1)/(q^r -q^r-1) to r, depending on R     

A Framework for the Construction ofGolay Sequences

In 1999, Davis and Jedwab gave an explicit algebraic normal form for m!/2 - 2^h(m+1) ordered Golay pairs of length 2^mmiddot over Z₂h, involving m!/2 - 2^h(m+1) Golay sequences. In 2005, Li and Chu unexpectedly found an additional 1024 length 16 quaternary Golay sequences. Fiedler and Jedwab showed in 2006 that these new Golay sequences exist because of a "crossover" of the aperiodic autocorrelation function of certain quaternary length eight sequences belonging to Golay pairs, and that they spawn further new quaternary Golay sequences and pairs of length 2^m for m > 4 under Budisin's 1990 iterative construction. The total number of Golay sequences and pairs spawned in this way is counted, and their algebraic normal form is given explicitly. A framework of constructions is derived in which Turyn's 1974 product construction, together with several variations, plays a key role. All previously known Golay sequences and pairs of length 2^m over Z₂h can be obtained directly in explicit algebraic normal form from this framework. Furthermore, additional quaternary Golay sequences and pairs of length 2^m are produced that cannot be obtained from any other known construction. The framework generalizes readily to lengths that are not a power of 2, and to alphabets other than Z_2h .     

Stopping Set Analysis of Iterative Row-Column Decoding of Product Codes

In this paper, we introduce stopping sets for iterative row-column decoding of product codes using optimal constituent decoders. When transmitting over the binary erasure channel (BEC), iterative row-column decoding of product codes using optimal constituent decoders will either be successful, or stop in the unique maximum-size stopping set that is contained in the (initial) set of erased positions. Let C_p denote the product code of two binary linear codes C_c and C_r of minimum distances dc and d_r and second generalized Hamming weights d₂(C_c) and d₂(C_r), respectively. We show that the size s_min of the smallest noncode- word stopping set is at least mm(d_rd₂(C_c),d_cd₂(C_r)) > d_rd_c, where the inequality follows from the Griesmer bound. If there are no codewords in C_p with support set S, where S is a stopping set, then S is said to be a noncodeword stopping set. An immediate consequence is that the erasure probability after iterative row-column decoding using optimal constituent decoders of (finite-length) product codes on the BEC, approaches the erasure probability after maximum-likelihood decoding as the channel erasure probability decreases. We also give an explicit formula for the number of noncodeword stopping sets of size s_min, which depends only on the first nonzero coefficient of the constituent (row and column) first and second support weight enumerators, for the case when d₂(C_r) < 2d_r and d₂(C_c) < 2d_c. Finally, as an example, we apply the derived results to the product of two (extended) Hamming codes and two Golay codes.     

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     

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.     

关于BCH码的广义Hamming重量上,下限

一个线性码的第ｒ广义Ｈａｍｍｉｎｇ重量是它任意ｒ维子码的最小支集大小。本文给出了一般（本原、狭义）ＢＣＨ码的广义Ｈａｍｍｉｎｇ重量下限和一类ＢＣＨ码的广义Ｈａｍｍｉｎｇ重量上限     

Weight hierarchies of extremal non-chain binary codes of dimension4

The weight hierarchy of a linear [n,k;q] code C over GF(q) is the sequence (d₁,d₂,···,d_k ) where d_r is the smallest support of an r-dimensional subcode of C. An [n,k;q] code is extremal nonchain if, for any r and s, where 1⩽rS(D)=d_r, and w_S (E)=d_s. The possible weight hierarchies of such binary codes of dimension 4 are determined     

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     

On the Stopping Redundancy of Reed&#8211;Muller Codes

The stopping redundancy of the code is an important parameter which arises from analyzing the performance of a linear code under iterative decoding on a binary erasure channel. In this paper, we will consider the stopping redundancy of Reed-Muller codes and related codes. Let R(lscr,m) be the Reed-Muller code of length 2^m and order lscr. Schwartz and Vardy gave a recursive construction of parity-check matrices for the Reed-Muller codes, and asked whether the number of rows in those parity-check matrices is the stopping redundancy of the codes. We prove that the stopping redundancy of R(m-2,m), which is also the extended Hamming code of length 2^m, is 2m-1 and thus show that the recursive bound is tight in this case. We prove that the stopping redundancy of the simplex code equals its redundancy. Several constructions of codes for which the stopping redundancy equals the redundancy are discussed. We prove an upper bound on the stopping redundancy of R(1,m). This bound is better than the known recursive bound and thus gives a negative answer to the question of Schwartz and Vardy     

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     

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