On the Undetected Error Probability for Shortened Hamming Codes

Shortened Hamming codes are widely used for error detection in data communications. In this paper, a method for computing the probability of an undetected error for these codes is presented. This method is then used to evaluate the error-detection performance of the shortened codes obtained from the two distance-4 Hamming codes adopted by CCITT X.25 for error control for packet-switched networks. We show that shortening a code does affect its error-detection performance.     

关于某些BCH码的不可检错误概率

本文研究了二元对称信道上，某些ＢＣＨ码的不可检错误概率。首先得到了几个重要的不等式。然后证明了纠正两个错误二进制本原ＢＣＨ码的扩展码是最佳检错码。     

On the Probability of Undetected Error for Linear Block Codes

The problem of computing the probability of undetected error is considered for linear block codes used for error detection. The recent literature is first reviewed and several results are extended. It is pointed out that an exact calculation can be based on either the weight distribution of a code or its dual. Using the dual code formulation, the probability of undetected error for the ensemble of all nonbinary linear block codes is derived as well as a theorem that shows why the probability of undetected error may not be a monotonic function of channel error rate for some poor codes. Several bounds on the undetected error probability are then presented. We conclude with detailed examples of binary and nonbinary codes for which exact results can be obtained. An efficient technique for measuring an unknown weight distribution is suggested and exact results are compared with experimental results.     

关于等重码不可检错误概率的界

本文研究了二元等重码不可检错误概率(UEP)的界.首先,我们通过研究二元等重码的对偶距离分布及其性质,给出二元等重码UEP的一个新的下界,该下界改进了Fu-Kl  ve-Wei的最新结果;然后,我们指出2003年Fu-Kl  ve-Wei关于二元等重码UEP上界的某些结果有错误,我们随后给出更正后的结果,即二元等重码UEP的平均值和一个上界.     

On the Probability of Undetected Error for Overextended Reed–Solomon Codes

Upper and lower bounds on the weight distribution of overextended Reed-Solomon (OERS) codes are derived, from which tight upper and lower bounds on the probability of undetected error for OERS codes are obtained for q-ary symmetric channels     

Evaluation of the Undetected Error Probability of Single Parity-Check Product Codes

In this paper, an expression for the undetected error probability(Pepsilon)of single parity-check product (SPCP) codes used for error detection over a binary symmetric channel is derived. It is shown that square SPCP codes need not obey a certain commonly used bound. Approximate expressions for the maximum(Pepsilon)and the corresponding maximizing ε are given.     

End-to-End Probability of Undetected Error in Computer Networks

The occurrence of undetected error during packet transmission in a computer network can potentially lead to faulty network operations. Presented here is a Markov analysis of the probability of undetected error on an end-to end basis. A practical rule-of-thumb is also derived.     

Exact and Approximate Expressions for the Probability of Undetected Errors of Varshamov–Tenengol'ts Codes

Computation of the undetected error probability for error detecting codes over the Z-channel is an important issue, explored only in part in previous literature. In this paper, Varshamov–Tenengol'ts (VT) codes are considered. First, an exact formula for the probability of undetected errors is given. It can be explicitly computed for small code lengths (up to approximately $25$). Next, some lower bounds that can be explicitly computed up to almost twice this length are studied. A comparison to the Hamming codes is given. It is further shown that heuristic arguments give a very good approximation that can easily be computed even for large lengths. Finally, Monte Carlo methods are used to estimate performance for long code lengths.      

Error Probability for Convolutional Codes with Weight Enumerating Function

The paper introduces the state reduction algorithm and accelerated state reduction algorithm are used to compute the distance weight enumerator(transfer function) T[x,y] of convolutional codes.Next use computer simulation to compare upper bound on the bit error probability on an additive white Gaussian noise (AWGN) for maximum free distance (MFD) codes of previously found and optimum distance spectrum (ODS) codes with rate 1/4,overall constraint length are 5 and 7,respectively. Finally,a method of how to search for good convolutional codes is given.     

基于最大秩距离码的McEliece公钥加密方案

提出了基于最大秩距离码新的McElience公钥密码系统，讨论其可行性及安全性．证明了它比基于纠错码的McElience公钥密码系统里安全。而且，通过引入单向函数．使密码系统能有效地抗击消息重发攻击和相关消患攻击。     

基于最大秩距离码的数字签名方案

G.Kabatianskii等在<基于随机纠错码的数字签名方案>一文中,基于随机纠错码构造了一数字签名方案.本文依据上述文章中的构造方法,基于最大秩距离码,构造一新的数字签名方案,说明了该方案比G.Kabatianskii等的方案更实用.     

关于等重码的对偶距离

本文借助于对偶距离和相关免疫阶的关系，证明了等重码的对偶距离为１或２。     

极低误码率下Turbo码的性能估计

Turbo码在中低信噪比下具有接近信道容量极限的优异性能,但在误码率极低、信噪比较高下性能曲线下降平缓,并且难以进行大计算量的仿真。使用输入信息序列片段长度递增筛选法,逐步筛选出所有产生自由距码字的信息帧,从而得出Turbo码完整的自由距特性,并用来估计极低误码率下的性能,该方法有效地减小了完整枚举法的计算量,且保证了精确性。     

等距码的对偶距离分布及其性质

本文主要讨论了等距码的对偶距离分布及其性质，然后利用这些性质将［１］中的某些结果推广到ｑ元等距码情形，并得到了其对偶距离分布的递推关系式，最后，本文给出了ｑ元等距码的码字数目的一个上界。     

On the Minimal Distance of Binary Self-Dual Cyclic Codes

In this paper, an explicit construction of binary self-dual cyclic codes of length n going to infinity with a minimal distance at least half the square root of n is presented. The same idea is also used to construct more general binary cyclic codes with a large minimal distance. Finally, in the special case of self-dual cyclic codes, a simplified version of a proof by Conway and Sloane is given, showing an upper bound for the distance of binary self-dual codes.     

Tightened Upper Bounds on the ML Decoding Error Probability of Binary Linear Block Codes

The performance of maximum-likelihood (ML) decoded binary linear block codes is addressed via the derivation of tightened upper bounds on their decoding error probability. The upper bounds on the block and bit error probabilities are valid for any memoryless, binary-input and output-symmetric communication channel, and their effectiveness is exemplified for various ensembles of turbo-like codes over the additive white Gaussian noise (AWGN) channel. An expurgation of the distance spectrum of binary linear block codes further tightens the resulting upper bounds     

An Improved Bound on the List Error Probability and List Distance Properties

List decoding of binary block codes for the additive white Gaussian noise (AWGN) channel is considered. The output of a list decoder is a list of the most likely codewords, that is, the signal points closest to the received signal in the Euclidean-metric sense. A decoding error occurs when the transmitted codeword is not on this list. It is shown that the list error probability is fully described by the so-called list configuration matrix, which is the Gram matrix obtained from the signal vectors forming the list. The worst case list configuration matrix determines the minimum list distance of the code, which is a generalization of the minimum distance to the case of list decoding. Some properties of the list configuration matrix are studied and their connections to the list distance are established. These results are further exploited to obtain a new upper bound on the list error probability, which is tighter than the previously known bounds. This bound is derived by combining the techniques for obtaining the tangential union bound with an improved bound on the error probability for a given list. The results are illustrated by examples.     

再论二元非线性等重码的检错性能

本文利用对偶距离分布的性质,从理论上证明了文「４」的必要条件在δ＝１时实际上是充分条件,并得出对于（ｎ,２,ω）码,最佳检错码与检错好码是等价的,最后,本文讨论了（ｎ,２,ω）码不可检错误概率的界。     

关于二元等重码的最大码字数

本文利用Johnson Schemes理论研究了二元等重码及其最大码字数问题.在Delsarte的associate schemes理论中,Q-变换被引入以研究二元等重码的距离分布.首先,本文研究了等重码距离分布的Q-变换;然后,通过使用Q-变换的性质,我们研究了二元等重码的最大码字数问题并得到码字数的一个新的上界,该上界在形式上类似于纠错码理论中的Grey-Rankin界,并且在某些情况下优于已知的结果.