Youcef Maouche | Searcharxiv

arXiv2017

Two-Weight and a Few Weights Trace Codes over $\mathbb{F}_{q}+u\mathbb{F}_{q}$

Let $p$ be a prime number, $q=p^s$ for a positive integer $s$. For any positive divisor $e$ of $q-1$, we construct an infinite family codes of size $q^{2m}$ with few Lee-weight. These codes are defined as trace codes over the ring $R=\mathbb{F}_q + u\mathbb{F}_q$, $u^2 = 0$. Using Gauss sums, their Lee weight distributions are provided. When $\gcd(e,m)=1$, we obtain an infinite family of two-weight codes over the finite field $\mathbb{F}_q$ which meet the Griesmer bound. Moreover, when $\gcd(e,m)=2, 3$ or $4$ we construct new infinite family codes with at most five-weight.

cs.IT↗

arXiv2017

Some Repeated-Root Constacyclic Codes over Galois Rings

Codes over Galois rings have been studied extensively during the last three decades. Negacyclic codes over $GR(2^a,m)$ of length $2^s$ have been characterized: the ring $\mathcal{R}_2(a,m,-1)= \frac{GR(2^a,m)[x]}{\langle x^{2^s}+1\rangle}$ is a chain ring. Furthermore, these results have been generalized to $λ$-constacyclic codes for any unit $λ$ of the form $4z-1$, $z\in GR(2^a, m)$. In this paper, we study more general cases and investigate all cases where $\mathcal{R}_p(a,m,γ)= \frac{GR(p^a,m)[x]}{\langle x^{p^s}-γ\rangle}$ is a chain ring. In particular, necessary and sufficient conditions for the ring $\mathcal{R}_p(a,m,γ)$ to be a chain ring are obtained. In addition, by using this structure we investigate all $γ$-constacyclic codes over $GR(p^a,m)$ when $\mathcal{R}_p(a,m,γ)$ is a chain ring. Necessary and sufficient conditions for the existence of self-orthogonal and self-dual $γ$-constacyclic codes are also provided. Among others, for any prime $p$, the structure of $\mathcal{R}_p(a,m,γ)=\frac{GR(p^a,m)[x]}{\langle x^{p^s}-γ\rangle}$ is used to establish the Hamming and homogeneous distances of $γ$-constacyclic codes.

cs.IT↗