An explicit class of codes with good parameters and their duals

We study a class of codes with good parameters and their duals explicitly. We give direct constructions of the dual codes and obtain self-orthogonal codes with good parameters.     

Chinese remainder codes

Chinese remainder codes are constructed by applying weak block designs and the Chinese remainder theorem of ring theory. The new type of linear codes take the congruence class in the congruence class ring R/I ₁ ∩ I ₂ ∩ ··· ∩ I _n for the information bit, embed R/J _i into R/I ₁ ∩ I ₂ ∩ ··· ∩ I _n, and assign the cosets of R/J _i as the subring of R/I ₁ ∩ I ₂ ∩ ··· ∩ I _n and the cosets of R/J _i in R/I ₁ ∩ I ₂ ∩ ··· ∩ I _n as check lines. Many code classes exist in the Chinese remainder codes that have high code rates. Chinese remainder codes are the essential generalization of Sun Zi codes. Selected from Journal of Mathematical Research and Exposition, 2004, 24(2): 347–352     

二元非线性2—重量码的检错性能

设Ｃ为二元非线性２－重量码（ｎ，２，ｗ１，ｗ２），本文讨论了Ｃ的不可检错误概率，首先给出了Ｃ成为最佳检错码的一个必要条件，接着我们证明了当ｎ〉２ｗ２时，Ｃ一定是不是最佳检错码，然后，对于ｗ１＋ｗ２＝ｎ的情形，我们找出了其中所有的最佳检错误。     

选择式频率分集SFH／MFSK系统在短波信道中的性能

研究了选择式频率分集Ｍ进制频移键控慢跳频（ＳＦＨ／ＭＦＳＫ）系统在多径瑞利衰落信道中的性能，分析了部分边带干扰，多径干扰，信道热噪声和调制阶数Ｍ对无编码和有编码系统性能的影响．研究结果表明，多径干扰强度和信道热噪声对系统性能影响较大，分集仅能在一定程度上削弱它们对系统性能的影响，只有在分集与编码相结合，并采用适当的调制阶数Ｍ时，才能显著地改善系统的性能．     

The New Minimum Distance Bounds of Goppa Codes and Their Decoding

A couple of new lower bounds of the minimum distance of Goppa codes is derived, using an extended field code for a Goppa code which contains the Goppa code as its subfield-subcode. Also presented are procedures for both error-only and error-and-erasure decoding for Goppa codes up to the new lower bounds, based on the Berlekamp-Massey algorithm and the Feng-Tzeng multisequence shift-register synthesis algorithms which have been used for decoding cyclic codes up to the BCH and HT(Hartmann-Tzeng) bounds.     

Fast Decoding Algorithms for First Order Reed-Muller and Related Codes

Fast decoding algorithms for short codes based on modifications of maximum likelihood decoding algorithms of first order Reed-Muller codes are described. Only additions-subtractions, comparisons and absolute value calculations are used in the algorithms. Soft and hard decisions maximum likelihood decoding algorithms for first order Reed-Muller and the Nordstrom-Robinson codes with low complexity are proposed.     

A Subclass of Binary Goppa Codes With Improved Estimation of the Code Dimension

A new bound for the dimension of binary Goppa codes belonging to a specific subclass is given. This bound improves the well-known lower bound for Goppa codes.     

Bounds on Spectra of Codes with Known Dual Distance

We estimate the interval where the distance distribution of a code of length n and of given dual distance is upperbounded by the binomial distribution. The binomial upper bound is shown to be sharp in this range in the sense that for every subinterval of size about √n ln n there exists a spectrum component asymptotically achieving the binomial bound. For self-dual codes we give a better estimate for the interval of binomiality.     

Implementation of LT codes based on chaos

Fountain codes provide an efficient way to transfer information over erasure channels like the Internet. LT codes are the first codes fully realizing the digital fountain concept. They are asymptotically optimal rateless erasure codes with highly efficient encoding and decoding algorithms. In theory, for each encoding symbol of LT codes, its degree is randomly chosen according to a predetermined degree distribution, and its neighbours used to generate that encoding symbol are chosen uniformly at random. Practical implementation of LT codes usually realizes the randomness through pseudo-randomness number generator like linear congruential method. This paper applies the pseudo-randomness of chaotic sequence in the implementation of LT codes. Two Kent chaotic maps are used to determine the degree and neighbour（s） of each encoding symbol. It is shown that the implemented LT codes based on chaos perform better than the LT codes implemented by the traditional pseudo-randomness number generator.     

Two-Weight Codes, Partial Geometries and Steiner Systems

Two-weight codes and projectivesets having two intersection sizes with hyperplanes are equivalentobjects and they define strongly regular graphs. We construct projective sets in PG(2m – 1,q) that have the sameintersection numbers with hyperplanes as the hyperbolic quadricQ⁺(2m – 1,q). We investigate these sets; we provethat if q = 2 the corresponding strongly regular graphsare switching equivalent and that they contain subconstituentsthat are point graphs of partial geometries. If m = 4the partial geometries have parameters s = 7, t = 8, = 4 and some of them are embeddable in Steinersystems S(2,8,120).