arXiv subjects

Sucheta Dutt

Publications and source records attributed to Sucheta Dutt.

9 recordsLinked to original sources

arXiv2026

Two-dimensional constacyclic codes over finite chain rings

The main focus of this paper is on the algebraic structure of two-dimensional $(\lambda,\mu)$-constacyclic codes of length $\ell\mathrm{m}$ over finite chain rings with residue field $\mathbb{F}_q$, where $q \equiv 1 \pmod{r\mathrm{m}}$ and $r$ denotes the multiplicative order of $\bar{\mu}$. In this paper, the structure of two-dimensional $(\lambda,\mu)$-constacyclic codes is obtained. Our approach relies on analysing primitive idempotents within the finite chain ring to determine the generators of these codes. We also find the condition under which two-dimensional constacyclic codes are maximum Hamming distance with respect to rank (MHDR) over finite chain rings.

cs.IT↗

arXiv2026

On the structure of constacyclic codes over finite chain rings

In the present paper, we provide an explicit construction for generators of a $\lambda$-constacyclic code $\mathcal{C}$ of arbitrary length $\ell$ over a finite chain ring(FCR) $\mathcal{R}$ in terms of certain minimum degree polynomials of the ring $\mathcal{R}[x]/ \langle x^{\ell}-\lambda \rangle$. Moreover, the proposed construction achieves the minimum possible number of generators. We prove certain properties of this set of generators, using which we obtain a minimal spanning set of $\mathcal{C}$. We also obtain that the rank of $\mathcal{C}$ is $\ell-n_0$, where $n_0$ is the degree of the minimal degree polynomial in $\mathcal{C}$. Finally, we derive necessary and sufficient conditions under which an arbitrary length $\lambda$-constacyclic code $\mathcal{C}$ over $\mathcal{R}$ is Maximum Hamming Distance with respect to Rank(MHDR) as well as Maximum Distance Separable(MDS) in terms of a torsion code of $\mathcal{C}$ over the residue field $\mathbb{F}_q$ of $\mathcal{R}$. We further determine the exact values for $n_0$ for which $\mathcal{C}$ over $\mathcal{R}$ is MHDR.

cs.IT↗

arXiv2024

On constacyclic codes over a class of non-chain rings

In this paper, a unique form of generators of a constacyclic code of arbitrary length over a non-chain ring of the type $\mathtt{R_{_{\theta}}}=Z_{4}+\nu Z_{4}, \nu^{2}=\theta \in Z_{4}+\nu Z_{4}$ has been obtained. Further, rank and cardinality of a constacyclic code of arbitrary length over a non-chain ring of the type $\mathtt{R_{_{\theta}}}$ have been obtained by determining a minimal spanning set of the code. Also, necessary and sufficient conditions for a constacyclic code of arbitrary length over a non-chain ring of the type $\mathtt{R_{_{\theta}}}$ to be reversible have been determined. Examples have also been presented in support of our results.

math.AC↗

arXiv2023

Reversible complement cyclic codes over finite chain rings

Let k be an arbitrary element of a finite commutative chain ring R and u be a unit in R. In this work, we present necessary conditions which are sufficient as well for a cyclic code to be a (u,k) reversible complement code over R. Using these conditions, all principally generated cyclic codes over the ring Z_{2}+vZ_{2}+v^{2}Z_{2}, v^{3}=0 of length 4 have been checked to find whether they are (1,1) reversible complement or not.

cs.IT↗

arXiv2023

Reversible cyclic codes over finite chain rings

In this paper, necessary and sufficient conditions for the reversibility of a cyclic code of arbitrary length over a finite commutative chain ring have been derived. MDS reversible cyclic codes having length p^s over a finite chain ring with nilpotency index 2 have been characterized and a few examples of MDS reversible cyclic codes have been presented. Further, it is shown that the torsion codes of a reversible cyclic code over a finite chain ring are reversible. Also, an example of a non-reversible cyclic code for which all its torsion codes are reversible has been presented to show that the converse of this statement is not true. The cardinality and Hamming distance of a cyclic code over a finite commutative chain ring have also been determined.

cs.IT↗

arXiv2023

Reversible and Reversible Complement Cyclic codes over a class of non-chain rings

In this paper, necessary and sufficient conditions for a cyclic code of arbitrary length over the non-chain rings $Z_{4}+\nu Z_{4}$ for $\nu^{2} \in \{0,1,\nu,2\nu,3\nu,2+\nu,2+3\nu,3+2\nu\}$ to be a reversible cyclic code have been established. Also, conditions for a cyclic code over these non-chain rings to be a reversible complement cyclic code which are necessary as well as sufficient have been determined. Some examples of reversible and reversible complement cyclic codes over these rings have also been presented.

cs.IT↗

arXiv2023

MDS and MHDR cyclic codes over finite chain rings

In this work, a unique set of generators for a cyclic code over a finite chain ring has been established. The minimal spanning set and rank of the code have also been determined. Further, sufficient as well as necessary conditions for a cyclic code to be an MDS code and for a cyclic code to be an MHDR code have been obtained. Some examples of optimal cyclic codes have also been presented.

cs.IT↗

arXiv2023

Structure and Rank of Cyclic codes over a class of non-chain rings

The rings $Z_{4}+\nu Z_{4}$ have been classified into chain rings and non-chain rings on the basis of the values of $\nu^{2} \in Z_{4}+\nu Z_{4}.$ In this paper, the structure of cyclic codes of arbitrary length over the rings $Z_{4}+\nu Z_{4}$ for those values of $\nu^{2}$ for which these are non-chain rings has been established. A unique form of generators of these codes has also been obtained. Further, rank and cardinality of these codes have been established by finding minimal spanning sets for these codes.

cs.IT↗

arXiv2023

An Algorithm to find the Generators of Multidimensional Cyclic Codes over a Finite Chain Ring

The aim of this paper is to determine the algebraic structure of multidimensional cyclic codes over a finite chain ring $\mathfrak{R}$. An algorithm to find the generator polynomials of $n$ dimensional ($n$D) cyclic codes of length $m_{1}m_{2}\dots m_{n}$ over $\mathfrak{R}$ has been developed using the generator polynomials of cyclic codes over $\mathfrak{R}$. Additionally, the generators of $n$D cyclic codes with length $m_{1}m_{2}\dots m_{n}$ over $\mathfrak{R}$ have been obtained as separable polynomials for the case $q\equiv 1(mod~ m_{j}), j\geq 2$, where $q=p^{r}$ is the cardinality of residue field of $\mathfrak{R}$.

cs.IT↗