On Hermitian LCD codes and their Gray image

We provide methods and algorithms to construct Hermitian linear complementary dual (LCD) codes over finite fields. We study existence of self-dual basis with respect to Hermitian inner product, and as an application, we construct Euclidean LCD codes by projecting the Hermitian codes over such a basis. Many optimal quaternary Hermitian and ternary Euclidean LCD codes are obtained. Comparisons with classical constructions are made.     

On LCD,self dual and isodual cyclic codes over finite chain rings

In this paper, LCD cyclic, self dual and isodual codes over finite chain rings are investigated. It was proven recently that a non-free LCD cyclic code does not exist over finite chain rings. Based on algebraic number theory, we introduce necessary and sufficient conditions for which all free cyclic codes over a finite chain ring are LCD. We have also obtained conditions on the existence of non trivial self dual cyclic codes of any length when the nilpotency index of the maximal ideal of a finite chain ring is even. Further, several constructions of isodual codes are given based on the factorization of the polynomial $x^n-1$ over a finite chain ring.     

Galois LCD codes over mixed alphabets

In this work we give a characterization of Galois Linear Complementary Dual codes and Galois-invariant codes over mixed alphabets of finite chain rings, which leads to the study of the Gray image of ${\mathbb{F}}_p{\mathbb{F}}_p[\theta ]$-linear codes, where $p\in \{2;3\}$ and $\theta \ne {\theta }^2=0$ that provides LCD codes over ${\mathbb{F}}_p$.     

Constructions of optimal LCD codes over large finite fields

In this paper¹, we prove existence of optimal complementary dual codes (LCD codes) over large finite fields. We also give methods to generate orthogonal matrices over finite fields and then apply them to construct LCD codes. Construction methods include random sampling in the orthogonal group, code extension, matrix product codes and projection over a self-dual basis.     

Z_2Z_4-加性负循环码的对偶

在Z_2Z_4-加性码的基础上研究其循环码,进一步地引入其负循环码.通过建立Z_2Z_4下的正交关系,得出其对偶仍是一个Z_2Z_4负循环码;通过在Z_2Z_4码与Z_4[x]-子模之间建立同构映射来刻画其负循环码的结构以及码的参数类型,并用构造性的方法推出了其对偶的最小生成集.这些结果,便于码元等参数的计算及其应用.     

The dual of a constacyclic code,self dual,reversible constacyclic codes and constacyclic codes with complementary duals over finite local Frobenius non-chain rings with nilpotency index 3

Over finite local Frobenius non-chain rings with nilpotency index 3 and when the length of the codes is relatively prime to the characteristic of the residue field of the ring, the structure of the dual of $\gamma$-constacyclic codes is established and the algebraic characterization of self-dual $\gamma$-constacyclic codes, reversible $\gamma$-constacyclic codes and $\gamma$-constacyclic codes with complementary dual are given. Generators for the dual code are obtained from those of the original constacyclic code.     

Equivalences among Z2s-linear Hadamard codes

The ${\mathbb{Z}}_{2^s}$-additive codes are subgroups of ${\mathbb{Z}}_{2^s}^n$, and can be seen as a generalization of linear codes over ${\mathbb{Z}}_2$ and ${\mathbb{Z}}_4$. A ${\mathbb{Z}}_{2^s}$-linear Hadamard code is a binary Hadamard code which is the Gray map image of a ${\mathbb{Z}}_{2^s}$-additive code. A partial classification of these codes by using the dimension of the kernel is known. In this paper, we establish that some ${\mathbb{Z}}_{2^s}$-linear Hadamard codes of length $2^t$ are equivalent, once $t$ is fixed. This allows us to improve the known upper bounds for the number of such nonequivalent codes. Moreover, up to $t=11$, this new upper bound coincides with a known lower bound (based on the rank and dimension of the kernel). Finally, when we focus on $s\in \{2,3\}$, the full classification of the ${\mathbb{Z}}_{2^s}$-linear Hadamard codes of length $2^t$ is established by giving the exact number of such codes.     

On the classification of linear complementary dual codes

We give a complete classification of binary linear complementary dual codes of lengths up to 13 and ternary linear complementary dual codes of lengths up to 10.     

On binary reflected Gray codes and functions

The binary reflected Gray code function b is defined as follows: If m is a nonnegative integer, then b(m) is the integer obtained when initial zeros are omitted from the binary reflected Gray code of m.This paper examines this Gray code function and its inverse and gives simple algorithms to generate both. It also simplifies Conder's result that the jth letter of the kth word of the binary reflected Gray code of length n is