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Register a new userDo non-free LCD codes over finite commutative Frobenius rings exist?
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Added by
Edgar Martinez-Moro
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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No Arabic abstract
In this paper, we clarify some aspects on LCD codes in the literature. We first prove that a non-free LCD code does not exist over finite commutative Frobenius local rings. We then obtain a necessary and sufficient condition for the existence of LCD code over finite commutative Frobenius rings. We later show that a free constacyclic code over finite chain ring is LCD if and only if it is reversible, and also provide a necessary and sufficient condition for a constacyclic code to be reversible over finite chain rings. We illustrate the minimum Lee-distance of LCD codes over some finite commutative chain rings and demonstrate the results with examples. We also got some new optimal $mathbb{Z}_4$ codes of different lengths {which are} cyclic LCD codes over $mathbb{Z}_4$.
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In this paper, we show that LCD codes are not equivalent to linear codes over small finite fields. The enumeration of binary optimal LCD codes is obtained. We also get the exact value of LD$(n,2)$ over $mathbb{F}_3$ and $mathbb{F}_4$. We study the bound of LCD codes over $mathbb{F}_q$.
In this paper we give the generalization of lifted codes over any finite chain ring. This has been done by using the construction of finite chain rings from $p$-adic fields. Further we propose a lattice construction from linear codes over finite chain rings using lifted codes.
Given a commutative ring $R$ with identity, a matrix $Ain M_{stimes l}(R)$, and $R$-linear codes $mathcal{C}_1, dots, mathcal{C}_s$ of the same length, this article considers the hull of the matrix-product codes $[mathcal{C}_1 dots mathcal{C}_s],A$. Consequently, it introduces various sufficient conditions under which $[mathcal{C}_1 dots mathcal{C}_s],A$ is a linear complementary dual (LCD) code. As an application, LCD matrix-product codes arising from torsion codes over finite chain rings are considered. Highlighting examples are also given.
Given $texttt{S}|texttt{R}$ a finite Galois extension of finite chain rings and $mathcal{B}$ an $texttt{S}$-linear code we define two Galois operators, the closure operator and the interior operator. We proof that a linear code is Galois invariant if and only if the row standard form of its generator matrix has all entries in the fixed ring by the Galois group and show a Galois correspondence in the class of $texttt{S}$-linear codes. As applications some improvements of upper and lower bounds for the rank of the restriction and trace code are given and some applications to $texttt{S}$-linear cyclic codes are shown.
Let $mathbb{F}_q$ be a finite field of order $q$, a prime power integer such that $q=et+1$ where $tgeq 1,egeq 2$ are integers. In this paper, we study cyclic codes of length $n$ over a non-chain ring $R_{e,q}=mathbb{F}_q[u]/langle u^e-1rangle$. We define a Gray map $varphi$ and obtain many { maximum-distance-separable} (MDS) and optimal $mathbb{F}_q$-linear codes from the Gray images of cyclic codes. Under certain conditions we determine { linear complementary dual} (LCD) codes of length $n$ when $gcd(n,q) eq 1$ and $gcd(n,q)= 1$, respectively. It is proved that { a} cyclic code $mathcal{C}$ of length $n$ is an LCD code if and only if its Gray image $varphi(mathcal{C})$ is an LCD code of length $4n$ over $mathbb{F}_q$. Among others, we present the conditions for existence of free and non-free LCD codes. Moreover, we obtain many optimal LCD codes as the Gray images of non-free LCD codes over $R_{e,q}$.
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