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     

On the generalized Hamming weights of several classes of cycliccods

The generalized Hamming weights of a linear code are fundamental code parameters related to the minimal overlap structures of the subcodes. They were introduced by V.K. Wei (1991) and shown to characterize the performance of the linear code in certain cryptographical applications. Results are presented on the generalized Hamming weights of several classes of binary cyclic codes, including primitive double-error-correcting and triple-error-correcting BCH codes, certain reversible cyclic codes, and some extended binary Goppa codes. In particular, the second generalized Hamming weight of primitive double-error-correcting BCH codes is determined and upper and lower bounds are obtained for the generalized Hamming weights for the codes studied. These bounds are compared to results from other methods     

广义Hamming重量上,下界的对偶定理

本文给出了一种广义Ｈａｍｍｉｎｇ重量上、下界的对偶定理。即若给定一个码的对偶码的广义Ｈａｍｍｉｎｇ重量上界（或者下界），可以给出该码的广义Ｈａｍｍｉｎｇ重量上界（或者下界）。Ｈ．Ｓｔｉｃｈ－ｎｏｔｈ（１９９４）曾给出了迹码（如ＢＣＨ码和Ｇｏｐｐａ码的对偶码）的广义Ｈａｍｍｉｎｇ重量一种上、下界，如果采用本文结果就可以给出迹码的对偶码的广义Ｈａｍｍｉｎｇ重量一种上、下界。因此，本文的结果是Ｈ．Ｓｔｉｃｈｎｏｔｈ的结果的有益补充     

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     

关于Goppa码、BCH码的广义Hamming重量

本文研究了Goppa码、BCH码的广义Hamming重量,给出了Goppa码的广义Hamming重量的一个下界以及求该下界的一个算法;对于本原、狭义BCH码,给出了后面一些广义Hamming重量的确切值。     

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     

Construction of some binary linear codes of minimum distance five

A method of constructing binary linear codes that have a minimum Hamming distance of five is presented. The author proposes a general formulation of the parity check matrix for the desired code and then derives necessary conditions for the code to have a minimum distance of five. Some new codes that satisfy the necessary conditions are described. Some efficient new codes are obtained. In particular, a (47,36,5) code is obtained that has six more information bits than the best previously known code with 11 check bits     

Footprints or generalized Bezout's theorem

In two previous papers, the first by Feng, Rao, Berg, and Zhu (see ibid., vol.43, p.1799-810, 1997) and the second by Feng, Zhu, Shi, and Rao (see Proc. 35th. Afferton Conf. Communication, Control and Computing, p.205-14, 1997), the authors use a generalization of Bezout's theorem to estimate the minimum distance and generalized Hamming weights for a class of error correcting codes obtained by evaluation of polynomials in points of an algebraic curve. The main aim of this article is to show that instead of using this rather complex method the same results and some improvements can be obtained by using the so-called footprint from Grobner basis theory. We also develop the theory further such that the minimum distance and the generalized Hamming weights not only can be estimated but also can actually be determined     

Improving the Upper Bounds on the Covering Radii of Binary Reed–Muller Codes

By deriving bounds on character sums of Boolean functions and by using the characterizations, due to Kasami , of those elements of the Reed-Muller codes whose Hamming weights are smaller than twice and a half the minimum distance, we derive an improved upper bound on the covering radius of the Reed-Muller code of order 2, and we deduce improved upper bounds on the covering radii of the Reed-Muller codes of higher orders     

Turbo乘积码构造方法的研究

Turbo乘积码因其具有接近香农限的译码性能和适合高速译码的并行结构,已成为纠错编码领域的研究热点。Turbo乘积码的分量码一般由扩展汉明码构造而成,所以该类码字编码和译码的硬件实现比较简单。当Turbo乘积码采用扩展汉明码作为子码时,随着信噪比的提高,码字的最小码重对误帧率的影响会逐步增大。文中改进了Turbo乘积码编码结构,在只增加较小的编译码复杂度和时延的情况下,提高了码字的最小码重,并减少了最小码重码字在码字空间所占的比例。通过仿真和分析,比较了这种码和TPC码在误帧率性能、码字的最小码重分布以及最小码间距估计上的差异。     

Parameters of Goppa codes revisited

We discuss parameters of Goppa (1970) codes, such as minimum distance, covering radius, distance distribution, and generalized Hamming weights. By a variation on the exponential sums method and combinatorial arguments, we sharpen known bounds on these parameters     

LDPC codes from generalized polygons

We use the theory of finite classical generalized polygons to derive and study low-density parity-check (LDPC) codes. The Tanner graph of a generalized polygon LDPC code is highly symmetric, inherits the diameter size of the parent generalized polygon, and has minimum (one half) diameter-to-girth ratio. We show formally that when the diameter is four or six or eight, all codewords have even Hamming weight. When the generalized polygon has in addition an equal number of points and lines, we see that the nonregular polygon based code construction has minimum distance that is higher at least by two in comparison with the dual regular polygon code of the same rate and length. A new minimum-distance bound is presented for codes from nonregular polygons of even diameter and equal number of points and lines. Finally, we prove that all codes derived from finite classical generalized quadrangles are quasi-cyclic and we give the explicit size of the circulant blocks in the parity-check matrix. Our simulation studies of several generalized polygon LDPC codes demonstrate powerful bit-error-rate (BER) performance when decoding is carried out via low-complexity variants of belief propagation.     

二进制序列的重只定理和Hamming距离定理

本文系统地研究了二进制序列的重量与互相关函数旁瓣之和的关系,Hamming距离与游程数之间的关系,证明了普适的重量定理和Hamming距离定理,在此基础上,进一步研究了标准正交变换,正交码、互补码在Tseng分解过程中的重量与Hamming距离之间的变换关系以及它们与游程之间的联系,获得了一系列重要的推论。     

Optimal self-dual codes over F/sub 2//spl times/F/sub 2/ with respect to the Hamming weight

In this paper, we study optimal self-dual codes and type IV self-dual codes over the ring F/sub 2//spl times/F/sub 2/ of order 4. We give improved upper bounds on minimum Hamming and Lee weights for such codes. Using the bounds, we determine the highest minimum Hamming and Lee weights for such codes of lengths up to 30. We also construct optimal self-dual codes and type IV self-dual codes.     

Euclidean distance for combinations of some Hamming codes and binary CPFSK modulation (Corresp.)

The integration of channel coding and modulation in a communication system to increase the Euclidean distance between modulated signals is analyzed. Systems using binary continuous-phase frequency-shift keying modulation and some block codes, such as Hamming codes and shortened Hamming codes, are considered. It is shown that the minimum Euclidean distance depends on the configuration of the parity-check matrixHof the code. For the examined codes the optimum configurations ofH, which give the maximum values of the minimum Euclidean distance, are determined.     

几种LDPC码的最小汉明距离的计算

本文提出一种计算LDPC码的真实最小汉明距离的方法．该方法能够用来计算多种LDPC码方案的真实最小汉明距离，比如准循环LDPC码、pi-旋转LDPC码等．该方法是通过计算码的环长间接地找到LDPC码最小距离，由于计算环长的计算量要远比直接计算最小汉明距离来得低，因而该算法能够在有限时间内找到LDPC码的真实最小距离．通过仿真表明，用目前主流的个人计算机利用该方法找出一个有最小距离24的码率为1／4的准循环LDPC码最小距离大概需要花77分钟。     

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     

视频字符叠加器

文章介绍了实现在电视信号上叠加字符的方法，并具体给出了一个实际用于水井电视检修的视频字符叠加器的软硬件。     

High-Rate Quasi-Cyclic Low-Density Parity-Check Codes Derived From Finite Affine Planes

This paper shows that several attractive classes of quasi-cyclic (QC) low-density parity-check (LDPC) codes can be obtained from affine planes over finite fields. One class of these consists of duals of one-generator QC codes. Presented here for codes contained in this class are the exact minimum distance and a lower bound on the multiplicity of the minimum-weight codewords. Further, it is shown that the minimum Hamming distance of a code in this class is equal to its minimum additive white Gaussian noise (AWGN) pseudoweight. Also discussed is a class consisting of codes from circulant permutation matrices, and an explicit formula for the rank of the parity-check matrix is presented for these codes. Additionally, it is shown that each of these codes can be identified with a code constructed from a constacyclic maximum distance separable code of dimension 2. The construction is similar to the derivation of Reed-Solomon (RS)-based LDPC codes presented by Chen and Djurdjevic Experimental results show that a number of high rate QC-LDPC codes with excellent error performance are contained in these classes     

广义Hamming重量和等重码

本文将线性码的广义Hamming重量的概念推广到非线性码上去,并导出了一种广义Elias界.对于线性等重码,本文给出了其完整的重量谱系.