arXiv subjects

Shikha Patel

Publications and source records attributed to Shikha Patel.

9 recordsLinked to original sources

arXiv2026

Construction of codes over a commutative non-unital ring from simplicial complexes and their applications

In this article, we investigate the construction of linear codes over a finite ring $\mathcal{S}$, where $\mathcal{S}$ is taken to be an extension of a commutative non-unital ring $I$ of order $p^2$. Our approach is based on the defining set method. The defining sets considered in this work are derived from general simplicial complexes that may contain multiple maximal elements. We determine the parameters of these codes over $\mathcal{S}$ and study their Gray images. We also study the corresponding subfield-like codes. We show that these Gray image codes and subfield-like codes produce several families of divisible codes. Furthermore, we establish sufficient conditions under which these codes are minimal, optimal, and self-orthogonal. As applications of our results, we obtain several families of projective few-weight codes, and locally recoverable codes with small locality. We also study the minimal access structures of secret-sharing schemes associated with the duals of these minimal codes. Moreover, we construct several families of strongly regular graphs from projective two-weight codes and determine their parameters explicitly.

cs.IT↗

arXiv2026

Construction of Cyclic Codes over a Class of Matrix Rings

Let $ \mathbb F_2[u]/ \langle u^k \rangle= \mathbb F_2+u\mathbb F_2+u^2\mathbb F_2+\cdots+u^{k-1}\mathbb F_2 ,$ where $u^k=0$ for a positive integer $k$, and $\mathcal{R}=M_4 (\mathbb F_2( u)/ \langle u^k \rangle)$ be the finite noncommutative non-chain matrix ring of order $4\times4$. This paper presents the construction of cyclic codes over the finite field $\mathbb F_{16}$ via the considered matrix ring $\mathcal{R}$. In this connection, first, we discuss the structure of the ring $\mathcal{R}$ and show that $\mathcal{R}$ is isomorphic to the ring $( \mathbb F_{16}+ v\mathbb F_{16} + v^2\mathbb F_{16} + v^3\mathbb F_{16}) + u(\mathbb F_{16} + v\mathbb F_{16} + v^2\mathbb F_{16} + v^3\mathbb F_{16}) + u^2(\mathbb F_{16} + v\mathbb F_{16} + v^2\mathbb F_{16}+ v^3\mathbb F_{16}) + \cdots + u^{k-1}(\mathbb F_{16} + v\mathbb F_{16} + v^2\mathbb F_{16} + v^3\mathbb F_{16})$ where $v^4=0, u^k=0, u^iv^j=v^ju^i$ for $i \in \{1,\dots, k-1\}$ and $j \in \{1, 2, 3\}$. Then, we establish the form of ideals of the ring $\mathcal{R}$ and related cyclic codes over $\mathcal{R}$. Further, we show that these cyclic codes can be written as the direct sums of $\mathcal{R}$-submodules of $\frac{\mathcal{R}[x]}{ }$, and derive the formula for the cardinality of cyclic codes over $\mathcal{R}$. Then, we consider the Euclidean and Hermitian duals of the derived cyclic codes over $\mathcal{R}$. Under the module isometry for $\mathcal{R}$, we use the Bachoc map and the Gray map, which takes a derived cyclic code over $\mathcal{R}$ to $\mathbb F_{16}$. Finally, we provide some non-trivial examples of linear codes over $\mathbb F_{16}$ with good parameters that support our derived results and compare a few codes with existing codes in the literature.

cs.IT↗

arXiv2025

Decoding Algorithms for Two-dimensional Constacyclic Codes over $\mathbb{F}_q$

We derive the spectral domain properties of two-dimensional (2-D) $(\lambda_1, \lambda_2)$-constacyclic codes over $\mathbb{F}_q$ using the 2-D finite field Fourier transform (FFFT). Based on the spectral nulls of 2-D $(\lambda_1, \lambda_2)$-constacyclic codes, we characterize the structure of 2-D constacyclic coded arrays. The proposed 2-D construction has flexible code rates and works for any code areas, be it odd or even area. We present an algorithm to detect the location of 2-D errors. Further, we also propose decoding algorithms for extracting the error values using both time and frequency domain properties by exploiting the sparsity that arises due to duality in the time and frequency domains. Through several illustrative examples, we demonstrate the working of the proposed decoding algorithms.

cs.IT↗

arXiv2024

On the Structure of Two-Dimensional Constacyclic Codes using Common Zero Sets

We consider two-dimensional $(\lambda_1, \lambda_2)$-constacyclic codes over $\mathbb{F}_{q}$ of area $M N$, where $q$ is some power of prime $p$ with $\gcd(M,p)=1$ and $\gcd(N,p)=1$. With the help of common zero (CZ) set, we characterize 2-D constacyclic codes. Further, we provide an algorithm to construct an ideal basis of these codes by using their essential common zero (ECZ) sets. We also describe the dual of 2-D constacyclic codes. Finally, we provide an encoding scheme for generating 2-D constacyclic codes from the generator tensor, implementable in a parallel fashion. Through examples, we illustrate that 2-D constacyclic codes can have better minimum distance compared to their cyclic counterparts with the same code area and code rate, generalizing prior work over 2-D binary cyclic coded arrays.

cs.IT↗

arXiv2024

Quantum Constacyclic BCH Codes over Qudits: A Spectral-Domain Approach

We characterize constacyclic codes in the spectral domain using the finite field Fourier transform (FFFT) and propose a reduced complexity method for the spectral-domain decoder. Further, we also consider repeated-root constacyclic codes and characterize them in terms of symmetric and asymmetric $q$-cyclotomic cosets. Using zero sets of classical self-orthogonal and dual-containing codes, we derive quantum error correcting codes (QECCs) for both constacyclic Bose-Chaudhuri-Hocquenghem (BCH) codes and repeated-root constacyclic codes. We provide some examples of QECCs derived from repeated-root constacyclic codes and show that constacyclic BCH codes are more efficient than repeated-root constacyclic codes. Finally, quantum encoders and decoders are also proposed in the transform domain for Calderbank-Shor-Steane CSS-based quantum codes. Since constacyclic codes are a generalization of cyclic codes with better minimum distance than cyclic codes with the same code parameters, the proposed results are practically useful.

quant-ph↗

arXiv2024

Encoding and Construction of Quantum Codes from $(\gamma,\Delta)$-cyclic Codes over a Class of Non-chain Rings

Let $\mathbb{F}_q$ be a finite field of $q=p^m$ elements where $p$ is a prime and $m$ is a positive integer. This paper considers $(\gamma,\Delta)$-cyclic codes over a class of finite non-chain commutative rings $\mathscr{R}_{q,s}=\mathbb{F}_q[v_1,v_2,\dots,v_s]/\langle v_i-v_i^2,v_iv_j=v_jv_i=0\rangle$ where $\gamma$ is an automorphism of $\mathscr{R}_{q,s}$, $\Delta$ is a $\gamma$-derivation of $\mathscr{R}_{q,s}$ and $1\leq i\neq j\leq s$ for a positive integer $s$. Here, we show that a $(\gamma,\Delta)$-cyclic code of length $n$ over $\mathscr{R}_{q,s}$ is the direct sum of $(\theta,\Im)$-cyclic codes of length $n$ over $\mathbb{F}_q$, where $\theta$ is an automorphism of $\mathbb{F}_q$ and $\Im$ is a $\theta$-derivation of $\mathbb{F}_q$. Further, necessary and sufficient conditions for both $(\gamma,\Delta)$-cyclic and $(\theta,\Im)$-cyclic codes to contain their Euclidean duals are established. Then, we obtain many quantum codes by applying the dual containing criterion on the Gray images of these codes. These codes have better parameters than those available in the literature. Finally, the encoding and error-correction procedures for our proposed quantum codes are discussed.

cs.IT↗

arXiv2023

$\mathbb{F}_q\mathcal{R}$-skew cyclic codes and their application to quantum codes

Let $p$ be a prime and $\mathbb{F}_q$ be the finite field of order $q=p^m$. In this paper, we study $\mathbb{F}_q\mathcal{R}$-skew cyclic codes where $\mathcal{R}=\mathbb{F}_q+u\mathbb{F}_q$ with $u^2=u$. To characterize $\mathbb{F}_q\mathcal{R}$-skew cyclic codes, we first establish their algebraic structure and then discuss the dual-containing properties by considering a non-degenerate inner product. Further, we define a Gray map over $\mathbb{F}_q\mathcal{R}$ and obtain their $\mathbb{F}_q$-Gray images. As an application, we apply the CSS (Calderbank-Shor-Steane) construction on Gray images of dual containing $\mathbb{F}_q\mathcal{R}$-skew cyclic codes and obtain many quantum codes with better parameters than the best-known codes available in the literature.

cs.IT↗

arXiv2019

Reversible cyclic codes over $\mathbb{F}_q + u \mathbb{F}_q$

Let $q$ be a power of a prime $p$. In this paper, we study reversible cyclic codes of arbitrary length over the ring $ R = \mathbb{F}_q + u \mathbb{F}_q$, where $u^2=0 mod q$. First, we find a unique set of generators for cyclic codes over $R$, followed by a classification of reversible cyclic codes with respect to their generators. Also, under certain conditions, it is shown that dual of reversible cyclic code is reversible over $\mathbb{Z}_2+u\mathbb{Z}_2$. Further, to show the importance of these results, some examples of reversible cyclic codes are provided.

cs.IT↗

arXiv2019

Skew Generalized Polycyclic Codes with Derivations

In this paper, we first consider the iterated skew polynomial ring $\mathscr{R}[z_1;\tau_1,\delta_{\tau_1}]$\\$[z_2;\tau_2,\delta_{\tau_2}]$, where $\mathscr{R}$ is a finite ring with unity. Then we use this structure for the construction of skew generalized polycyclic codes over the ring $\mathscr{R}$ and finite field $\mathbb{F}_q$, where $q=p^m$ for some positive integer $m$. Further, we derive the structure of the generator and parity check matrices for skew generalized polycyclic codes. Furthermore, we improve the Bose-Chaudhuri-Hocquenghem (BCH) lower bound for a minimum distance of skew generalized polycyclic codes with non-zero derivations over a finite field. Moreover, we find a sufficient condition for a code to be a maximum-distance-separable (MDS) code. In addition, we provide examples of MDS codes to show the importance of our results. A comparative summary of our work with other linear codes is also discussed.

cs.IT↗