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     

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     

Further results on generalized Hamming weights for Goethals andPreparata codes over Z4

This article contains results on the generalized Hamming weights (GHW) for the Goethals and Preparata codes over Z₄. We give an upper bound on the rth generalized Hamming weights d_r(m,j) for the Goethals code G_m(j) of length 2^m over Z ₄, when m is odd. We also determine d_3.5(m,j) exactly. The upper bound is shown to be tight up to r=3.5. Furthermore, we determine the rth generalized Hamming weight d_r(m) for the Preparata code of length 2^m over Z₄ when r=3.5 and r=4     

On covering radii and coset weight distributions of extremal binaryself-dual codes of length 56

We present a method to determine the complete coset weight distributions of doubly even binary self-dual extremal [56, 28, 12] codes. The most important steps are (1) to describe the shape of the basis for the linear space of rigid Jacobi polynomials associated with such codes in each index i, (2) to describe the basis polynomials for the coset weight enumerators of the assigned coset weight i by means of rigid Jacobi polynomials of index i. The multiplicity of the cosets of weight i have a connection with the frequency of the rigid reference binary vectors v of weight i for the Jacobi polynomials. This information is sufficient to determine the complete coset weight distributions. Determination of the covering radius of the codes is an immediate consequence of this method. One important practical advantage of this method is that it is enough to get information on 8190 codewords of weight 12 (minimal-weight words) in each such code for computing every necessary information     

Jacobi polynomials for singly even self-dual codes and the coveringradius problems

In this correspondence, we develop a method to determine the complete coset weight distributions of the class of singly even self-dual binary codes. Our basic tool is the Jacobi polynomials for the code. It describes and controls the coset weight enumerators. As the results of our present method, we give the complete coset weight distributions of some extremal singly even self-dual codes of lengths 14, 22, 32, 36, and 40, respectively. We give the generator matrices of the used codes of lengths 36 and 40, respectively     

Preparata codes through lattices

We define an elementary family of lattices, from which we obtain a family of extended cyclic codes with coefficients in the modular integers. The first nontrivial subfamily is the family of quaternary Preparata codes. The family of dual codes coincides with the extended low-correlation sequences introduced by Kumar, Helleseth, and Calderbank (1995)     

On testing for improper error detection codes

A family of tests for improper codes is given. These tests can be used in cases where the complete weight distribution of the code is unknown. It was found that knowledge of the number of minimum weight codewords can be used to greatly increase the effectiveness of the asymptotic Varshamov-Gilbert test. Further improvement is possible as more is known about the number of other weight codewords     

An infinite class of counterexamples to a conjecture concerning nonlinear resilient functions

The main construction for resilient functions uses linear errorcorrecting codes; a resilient function constructed in this way is said to be linear. It has been conjectured that if a resilient function exists, then a linear function with the same parameters exists. In this note we construct infinite classes of nonlinear resilient functions from the Kerdock and Preparata codes. We also show that linear resilient functions having the same parameters as the functions that we construct from the Kerdock codes do not exist. Thus, the aforementioned conjecture is disproved.kResearch supported by NSF Grant CCR-9121051.     

环Fp+vFp+v2Fp上线性码的各种重量计数器及其MacWilliams恒等式

首先给出了环R=F_p+vF_p+v²F_p上线性码及其对偶码的结构及其Gray象的性质.定义了环R上线性码的各种重量计数器并讨论了它们之间的关系,特别的,确定了该环上线性码与其对偶码之间关于完全重量计数器的MacWilliams恒等式,利用该恒等式,进一步建立了该环上线性码与其对偶码之间的一种对称形式的MacWilliams恒等式.最后,利用该对称形式的MacWilliams恒等式得到了该环上的Hamming重量计数器和Lee重量计数器的MacWilliams恒等式,利用不同的方法推广了文献[7]中的结果.     

Heuristic Algorithms for Constructing Binary Constant Weight Codes

Constant weight binary codes are used in a number of applications. Constructions based on mathematical structure are known for many codes. However, heuristic constructions unrelated to any mathematical structure can become of greater importance when the parameters of the code are larger. This paper considers the problem of finding constant weight codes with the maximum number of codewords from a purely algorithmic perspective. A set of heuristic and metaheuristic methods is presented and developed into a variable neighborhood search framework. The proposed method is applied to 383 previously studied cases with lengths between 29 and 63. For these cases it generates 153 new codes, with significantly increased numbers of codewords in comparison with existing constructions. For 10 of these new codes the number of codewords meets a known upper bound, and so these 10 codes are optimal. As well as the ability to generate new best codes, the approach has the advantage that it is a single method capable of addressing many sets of parameters in a uniform way.     

Design and analysis of codes with distance 4 and 6 minimizing the probability of decoder error

The problem of minimization of the decoder error probability is considered for shortened codes of dimension 2 ^m with distance 4 and 6. We prove that shortened Panchenko codes with distance 4 achieve the minimal probability of decoder error under special form of shortening. This shows that Hamming codes are not the best. In the paper, the rules for shortening Panchenko codes are defined and a combinatorial method to minimize the number of words of weight 4 and 5 is developed. There are obtained exact lower bounds on the probability of decoder error and the full solution of the problem of minimization of the decoder error probability for [39,32,4] and [72,64,4] codes. For shortened BCH codes with distance 6, upper and lower bounds on the number of minimal weight codewords are derived. There are constructed [45,32,6] and [79,64,6] BCH codes with the number of weight 6 codewords close to the lower bound and the decoder error probabilities are calculated for these codes. The results are intended for use in memory devices.     

多相正交序列的完全互补码的构造方法

在Suehiro提出的多相完全互补序列理论的基础上,对其进行改进,从而可以更容易地增加完全互补序列组的数量.所论述的方法扩展了完全互补序列组的数目,也提高了系统设计的灵活性.     

环Fp+uFp上的Kerdock码和Preparata码

 Kerdock码和Preparata码是两类著名的二元非线性码,它们比相同条件下的线性码含有更多的码字.Hammons等人在1994年发表的文献中证明了这两类码可视为环Z₄上循环码在Gray映射下的像,从而使得这两类码的编码和译码变得非常简单.环F₂+uF₂是介于环Z₄与域F₄之间的一种四元素环,因此分享了环Z₄与域F₄的一些好的性质,此环上的编码理论研究成为一个新的热点.本文首次将Kerdock码和Preparata码的概念引入到环F_p+uF_p上,证明了它们是一对对偶码;并给出Kerdock码的迹表示;当p=2时,建立了环F₂+uF₂上这两类码与域F₂上的Reed-Muller码之间的联系;并证明了二元一阶Reed-Muller码是环F₂+uF₂上Kerdock码的线性子码的Gray像.     

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)     

LDPC码的一种循环差集构造方法

提出了一种由组合数学中的循环差集构造LDPC码的新方法,它能产生大量的列重和行重均为恒定值的规则码,并且可以排除圈长为4的圈和减少圈长等于6的圈。利用和积译码算法通过计算机仿真验证了这种码字具有优良的特性。     

纠两个错的二元BCH码的代数完全译码

本文提出纠两个错的二元BCH码的代数完全译码方法。它实现起来比Hartmann的一步一步译码方法速度快,并且当对应校验子S1、S3的错误图样重量为3时,能找出所有对应同样校验子的重量为3的错误图样。同时,本文也建立了GF(2m)上三次方程在GF(2m)上有三个不同根的判别式,这在纠三个错的二元BCH码的完全译码中十分重要。     

RS码编码参数的盲识别

针对现有的RS(Reed-Solomon)码盲识别计算复杂度较大的问题,提出了一种新的识别方法.首先统计不同码长分组时的码重分布,并定义与理论码重分布之间的相似度系数,通过计算找出最相似的一组即对应正确的码长;然后建立二元假设,并确定判决门限对码根进行判定;通过遍历域内所有的本原多项式,找出完整的连续码根分布,进而完成生成多项式的识别.仿真结果表明,所提方法的计算量较其他方法明显减少,并能有效完成码长和生成多项式的识别,在误码率小于10-3时,对常用RS码的识别率能达到90％以上.     

An algebraic complete decoding for double-error-correcting binary BCH codes

An algebraic complete decoding for double-error-correcting binary BCH codes of primitive length is derived. The decoding is done more quickly than the step-by-step decoding devised by Hartmann. And if an error pattern corresponding with syndromess ₁ ands ₂ has weight 3, the decoding can find all error patterns of weight 3 corresponding with these syndromes. At the same time, a discriminant for a polynomial of degree 3 overGF(2 ^m ) has three distinct roots inGF2 ^m ) is also derived. The discriminant is very important for complete decoding of triple-error-correcting binary BCH codes.     

New results on serial concatenated and accumulated-convolutional turbo code performance

Previous methods for analyzing serial concatenated turbo codes employing union error bounds are extended to determine the complete output weight enumeration function of the code; this provides the opportunity to employ a more refined bound due to Polytrev, with considerably improved results limited, however, to block lengths of about 256 bits by computational constraints. The method is then applied to a new class of “accumulated-convolutional” codes, which is a simple special subclass of serial concatenated codes inspired by the “repeat-accumulate” codes of Divsalar et al. Performance appears to be superior to that of conventional codes and results are obtained for much longer block lengths, with impressive results in regions approaching channel capacity.     

New binary codes

In this paper constructions are given for combining two, three, or four codes to obtain new codes. The Andryanov-Saskovets construction is generalized. It is shown that the Preparata double-error-correcting codes may be extended by about (block length)^{1/2}symbols, of which only one is a check symbol, and thate-error-correcting BCH codes may sometimes be extended by (block !ength)^{1/e}symbols, of which only one is a check symbol. Several new families of linear and nonlinear double-error-correcting codes are obtained. Finally, an infinite family of linear codes is given withd/n = frac{1}{3}, the first three being the(24,2^12, 8)Golay code, a(48,2^15, 16)code, and a(96,2^18, 32)code. Most of the codes given have more codewords than any comparable code previously known to us.