Weight Enumerators of Double Circulant Codes and New Extremal Self-Dual Codes

In this paper it is shown that the weight enumerator of a bordered double circulant self-dual code can be obtained from those of a pure double circulant self-dual code and its shadow through a relationship between bordered and pure double circulant codes. As applications, a restriction on the weight enumerators of some extremal double circulant codes is determined and a uniqueness proof of extremal double circulant self-dual codes of length 46 is given. New extremal singly-even [44,22,8] double circulant codes are constructed. These codes have weight enumerators for which extremal codes were not previously known to exist.     

Structure Theorems for Group Ring Codes with an Application to Self-Dual Codes

Using ideas from the cohomology of finite groups, an isomorphism is established between a group ring and the direct sum of twisted group rings. This gives a decomposition of a group ring code into twisted group ring codes. In the abelian case the twisted group ring codes are (multi-dimensional) constacyclic codes. We use the decomposition to prove that, with respect to the Euclidean inner product, there are no self-dual group ring codes when the group is the direct product of a 2-group and a group of odd order, and the ring is a field of odd characteristic or a certain modular ring. In particular, there are no self-dual abelian codes over the rings indicated. Extensions of these results to non-Euclidean inner products are briefly discussed.     

On Harmonic Weight Enumerators of Binary Codes

We define some new polynomials associated to a linear binary code and a harmonic function of degree k. The case k=0 is the usual weight enumerator of the code. When divided by (xy) ^k , they satisfy a MacWilliams type equality. When applied to certain harmonic functions constructed from Hahn polynomials, they can compute some information on the intersection numbers of the code. As an application, we classify the extremal even formally self-dual codes of length 12.     

An Enumeration of Binary Self-Dual Codes of Length 32

A binary self-dual code of length 2k is a (2k, k) binary linear code C with the property that every pair of codewords in C are orthogonal. Two self-dual codes, C ₁ and C ₂, are equivalent if and only if there is a permutation of the coordinates of C ₁ that takes C ₁ into C ₂. The automorphism group of a binary code C is the set of all permutations of the coordinates of C that takes C into itself.The main topic of this paper is the enumeration of inequivalent binary self-dual codes. We have developed algorithms that will take lists of inequivalent small codes and produce lists of larger codes where each inequivalent code occurs only a few times. We have defined a canonical form for codes that allowed us to eliminate the overenumeration. So we have lists of inequivalent binary self-dual codes of length up to 32. The enumeration of the length 32 codes is new. Our algorithm also finds the size of the automorphism group so that we can compute the number of distinct binary self-dual codes for a specific length. This number can also be found by counting and matches our total.     

On the Covering Radius of Ternary Extremal Self-Dual Codes

In this paper, we investigate the covering radius of ternary extremal self-dual codes. The covering radii of all ternary extremal self-dual codes of lengths up to 20 were previously known. The complete coset weight distributions of the two inequivalent extremal self-dual codes of length 24 are determined. As a consequence, it is shown that every extremal ternary self-dual code of length up to 24 has covering radius which meets the Delsarte bound. The first example of a ternary extremal self-dual code with covering radius which does not meet the Delsarte bound is also found. It is worth mentioning that the found code is of length 32.     

On McEliece's Theorem

A new simple proof of the well-known theorem of McEliece about the complete weight enumerators of ternary self-dual codes is given.     

Weight Hierarchies of Linear Codes Satisfying the Chain Condition

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. By explicit construction, it is shown that if a sequence (a₁,a₂,...,a_k) satisfies certain conditions, then it is the weight hierarchy of a code satisfying the chain condition.     

Automorphisms of Constant Weight Codes and of Divisible Designs

We show that the automorphism group of a divisible design is isomorphic to a subgroup H of index 1 or 2 in the automorphism group of the associated constant weight code. Only in very special cases H is not the full automorphism group.     

关于线性方程组的一个注记

本文证明了方程组（In＋AB）x=0和（In＋BA）x=0解的个数是一致的。     

Constant Weight Codes and Group Divisible Designs

The study of a class of optimal constant weight codes over arbitrary alphabets was initiated by Etzion, who showed that such codes are equivalent to special GDDs known as generalized Steiner systems GS(t,k,n,g) Etzion. This paper presents new constructions for these systems in the case t=2, k=3. In particular, these constructions imply that the obvious necessary conditions on the length n of the code for the existence of an optimal weight 3, distance 3 code over an alphabet of arbitrary size are asymptotically sufficient.     

A Note on Invariants Over Finite Local Rings

In this note, we prove a conjecture of Steinberg [10 Steinberg , R. (1987). On Dickson's theorem on invariants. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34:699–707. [Google Scholar]] about modular invariants of some linear groups over finite commutative local rings.     

Weight distribution of some reducible cyclic codes

Let q=p^m where p is an odd prime, m3, k1 and gcd(k,m)=1. Let Tr be the trace mapping from to and . In this paper we determine the value distribution of following two kinds of exponential sums

and

where is the canonical additive character of . As an application, we determine the weight distribution of the cyclic codes and over with parity-check polynomial h₂(x)h₃(x) and h₁(x)h₂(x)h₃(x), respectively, where h₁(x), h₂(x) and h₃(x) are the minimal polynomials of π⁻¹, π⁻² and π^−(pk+1) over , respectively, for a primitive element π of .     

Linear codes with few weights from weakly regular plateaued functions

Linear codes with few nonzero weights have wide applications in secret sharing, authentication codes, association schemes and strongly regular graphs. Recently, Wu et al. (2020) obtained some few-weighted linear codes by employing bent functions. In this paper, inspired by Wu et al. and some pioneers' ideas, we use a kind of functions, namely, general weakly regular plateaued functions, to define the defining sets of linear codes. Then, by utilizing some cyclotomic techniques, we construct some linear codes with few weights and obtain their weight distributions. Notably, some of the obtained codes are almost optimal with respect to the Griesmer bound. Finally, we observe that our newly constructed codes are minimal for almost all cases.     

关于极大前缀码的刻划的一个注记

设X*是字母表X上的自由幺半群,以X*为顶点集构造一个语言图Γ(X*),引入语言图Γ(X*)的横截集的概念,给出了前缀码为极大前缀码的一个刻划,并推广了相关文献的结果.     

State Spaces of Convolutional Codes

There are various definitions of convolutional codes and each definition leads to a definition of code state space. In the usual definition of a convolutional code generated by a rational encoding matrix input sequences can be any Laurent series. It is proved that restricting input sequences to be rational functions or restricting output sequences to be finite do not change the code state space, and that restricting both input and output sequences to be finite may change the code state space.     

低密度奇偶校验码的构造

低密度奇偶校验码(LDPC)最早是由Gallager于1962年提出.它们是线性分组码,其比特错误率极大地接近香农界.1995年Mackay和Neal发掘了LDPC码的新应用后,LDPC码引起了人们的广泛关注.本文利用组合结构给出一些新的LDPC码:利用可分组设计构造一类Tanner图中不含四长圈的正则LDPC码.     

Self-Orthogonal 3-(56,12,65) Designs and Extremal Doubly-Even Self-Dual Codes of Length 56

In this paper, we show that the code generated by the rows of a block-point incidence matrix of a self-orthogonal 3-(56,12,65) design is a doubly-even self-dual code of length 56. As a consequence, it is shown that an extremal doubly-even self-dual code of length 56 is generated by the codewords of minimum weight. We also demonstrate that there are more than one thousand inequivalent extremal doubly-even self-dual [56,28,12] codes. This result shows that there are more than one thousand non-isomorphic self-orthogonal 3-(56,12,65) designs. AMS Classification: 94B05, 05B05     

On the Weight Hierarchy of Product Codes

Let V and W be codes and let C = V W be the product code of V and W. In [6] Wei and Yang, conjectured a formula for the generalized Hamming weights of V W in terms of those of V and W provided that both V and W satisfy the chain condition. Recently the conjecture has been proved by Schaathun [4]. In this paper we generalize the formula to a product code with more than two components.     

基于R-模的线性网络码

一般来讲,线性网络码是基于有限域的.本文的目的是要把基于有限域的线性网络码从理论上推广到基于一般R-模的线性网络码,其中R是一个给定的环.一类重要的R-模是Z-模,最简单的特例是整数环Z.另一大类重要的R-模是一般包含一个给定子环R的数域或代数数域的线性网络码.例如整数环Z(C)有理数域Q(C)实数域R(C)复数域C.推广后网络传送的消息(信号)可以是实数或复数,因此扩大了应用的领域.     

构作正交空间中的一类Cartesian认证码

利用正交几何构作出一类Cartesian认证码,并且计算了它们的参数.假定信源和编码规则都按等概率分布选取,求出了认证码的成功的模仿攻击概率PI和成功的替换攻击概率PS.本文构作的Cartesian认证码的成功的模仿攻击概率PI均达到了其下界.