Maouche Youcef | Searcharxiv

arXiv2023

Linear codes with arbitrary dimensional hull and pure LCD code

In this paper, we introduce a general construction of linear codes with small dimension hull from any non LCD codes. Furthermore, we show that for any linear code $\Co$ over $\F_q$ ($q > 3$) with $dim(Hull(\Co))=h$ there exist an equivalent codes $\Co_j$ with $dim(Hull(\Co_j))=j$ for any integer $0\leq j \leq h$. We also introduce the notion of pure LCD code; an LCD code and all its equivalent are LCD; and construct an infinite family of pure LCD codes. In addition, we introduce a general construction of linear codes with one dimension hull.

cs.IT↗

arXiv2016

A Note on Hamming distance of constacyclic codes of length $p^s$ over $\mathbb F_{p^m} + u\mathbb F_{p^m}$

For any prime $p$, $λ$-constacyclic codes of length $p^s$ over ${\cal R}=\mathbb{F}_{p^m} + u\mathbb{F}_{p^m}$ are precisely the ideals of the local ring ${\cal R}_λ=\frac{{\cal R}[x]}{\left\langle x^{p^s}-λ\right\rangle}$, where $u^2=0$. In this paper, we first investigate the Hamming distances of cyclic codes of length $p^s$ over ${\cal R}$. The minimum Hamming distances of all cyclic codes of length $p^s$ over ${\cal R}$ are determined. Moreover, an isometry between cyclic and $α$-constacyclic codes of length $p^s$ over ${\cal R}$ is established, where $α$ is a nonzero element of $\mathbb{F}_{p^m}$, which carries over the results regarding cyclic codes corresponding to $α$-constacyclic codes of length $p^s$ over ${\cal R}$.

cs.IT↗