Galois images of polycyclic codes over a finite chain ring $S$ and their annihilator dual are investigated. The case when a polycyclic codes is Galois-disjoint over the ring $S,$ is characterized and, the trace codes and restrictions of free polycyclic codes over $S$ are also determined givind an analogue of Delsarte theorem among trace map, any S -linear code and its annihilator dual.
A structure theorem of the group codes which are relative projective for the subgroup $lbrace 1 rbrace$ of $G$ is given. With this, we show that all such relative projective group codes in a fixed group algebra $RG$ are in bijection to the chains of projective group codes of length $ell$ in the group algebra $mathbb{F}G$, where $mathbb{F}$ is the residue field of $R$. We use a given chain to construct the dual code in $RG$ and also derive the minimum Hamming weight as well as a lower bound of the minimum euclidean weight.
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.
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.
In this paper, we explore some properties of Galois hulls of cyclic serial codes over a chain ring and we devise an algorithm for computing all the possible parameters of the Euclidean hulls of that codes. We also establish the average $p^r$-dimension of the Euclidean hull, where $mathbb{F}_{p^r}$ is the residue field of $R$, and we provide some results of its relative growth.
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 $R_{e,q}$.