arXiv subjects

Habibul Islam

Publications and source records attributed to Habibul Islam.

11 recordsLinked to original sources

arXiv2024

Galois hulls of constacyclic codes over affine algebra rings

Let $\mathcal A$ the affine algebra given by the ring $\mathbb{F}_q[X_1,X_2,\ldots,X_\ell]/ I$, where $I$ is the ideal $\langle t_1(X_1), t_2(X_2), \ldots, t_\ell(X_\ell) \rangle$ with each $t_i(X_i)$, $1\leq i\leq \ell$, being a square-free polynomial over $\mathbb{F}_q$. This paper studies the $k$-Galois hulls of $\lambda$-constacyclic codes over $\mathcal A$ regarding their idempotent generators. For this, first, we define the $k$-Galois inner product over $\mathcal A$ and find the form of the generators of the $k$-Galois dual and the $k$-Galois hull of a $\lambda$-constacyclic code over $\mathcal A$. Then, we derive a formula for the $k$-Galois hull dimension of a $\lambda$-constacyclic code. Further, we provide a condition for a $\lambda$-constacyclic code to be $k$-Galois LCD. Finally, we give some examples of the use of these codes in constructing entanglement-assisted quantum error-correcting codes.

cs.IT↗

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

Dihedral Quantum Codes

We establish dihedral quantum codes of short block length, a class of CSS codes obtained by the lifted product construction. We present the code construction and give a formula for the code dimension, depending on the two classical codes that the CSS code is based on. We also give a lower bound on the code distance and construct an example of short dihedral quantum codes.

quant-ph↗

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↗

arXiv2022

Galois Hull Dimensions of Gabidulin Codes

For a prime power $q$, an integer $m$ and $0\leq e\leq m-1$ we study the $e$-Galois hull dimension of Gabidulin codes $G_k(\boldsymbol{\alpha})$ of length $m$ and dimension $k$ over $\mathbb{F}_{q^m}$. Using a self-dual basis $\boldsymbol{\alpha}$ of $\mathbb{F}_{q^m}$ over $\mathbb{F}_q$, we first explicitly compute the hull dimension of $G_k(\boldsymbol{\alpha})$. Then a necessary and sufficient condition of $G_k(\boldsymbol{\alpha})$ to be linear complementary dual (LCD), self-orthogonal and self-dual will be provided. We prove the existence of $e$-Galois (where $e=\frac{m}{2}$) self-dual Gabidulin codes of length $m$ for even $q$, which is in contrast to the known fact that Euclidean self-dual Gabidulin codes do not exist for even $q$. As an application, we construct two classes of entangled-assisted quantum error-correcting codes (EAQECCs) whose parameters have more flexibility compared to known codes in this context.

cs.IT↗

arXiv2021

Cyclic codes over a non-chain ring $R_{e,q}$ and their application to LCD codes

Let $\mathbb{F}_q$ be a finite field of order $q$, a prime power integer such that $q=et+1$ where $t\geq 1,e\geq 2$ are integers. In this paper, we study cyclic codes of length $n$ over a non-chain ring $R_{e,q}=\mathbb{F}_q[u]/\langle u^e-1\rangle$. 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 $\gcd(n,q)\neq 1$ 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 $R_{e,q}$.

cs.IT↗

arXiv2019

Constacyclic codes of length $4p^s$ over the Galois ring $GR(p^a,m)$

For prime $p$, $GR(p^a,m)$ represents the Galois ring of order $p^{am}$ and characterise $p$, where $a$ is any positive integer. In this article, we study the Type (1) $\lambda$-constacyclic codes of length $4p^s$ over the ring $GR(p^a,m)$, where $\lambda=\xi_0+p\xi_1+p^2z$, $\xi_0,\xi_1\in T(p,m)$ are nonzero elements and $z\in GR(p^a,m)$. In first case, when $\lambda$ is a square, we show that any ideal of $\mathcal{R}_p(a,m,\lambda)=\frac{GR(p^a,m)[x]}{\langle x^{4p^s}-\lambda\rangle}$ is the direct sum of the ideals of $\frac{GR(p^a,m)[x]}{\langle x^{2p^s}-\delta\rangle}$ and $\frac{GR(p^a,m)[x]}{\langle x^{2p^s}+\delta\rangle}$. In second, when $\lambda$ is not a square, we show that $\mathcal{R}_p(a,m,\lambda)$ is a chain ring whose ideals are $\langle (x^4-\alpha)^i\rangle\subseteq \mathcal{R}_p(a,m,\lambda)$, for $0\leq i\leq ap^s$ where $\alpha^{p^s}=\xi_0$. Also, we prove the dual of the above code is $\langle (x^4-\alpha^{-1})^{ap^s-i}\rangle\subseteq \mathcal{R}_p(a,m,\lambda^{-1})$ and present the necessary and sufficient condition for these codes to be self-orthogonal and self-dual, respectively. Moreover, the Rosenbloom-Tsfasman (RT) distance, Hamming distance and weight distribution of Type (1) $\lambda$-constacyclic codes of length $4p^s$ are obtained when $\lambda$ is not a square.

cs.IT↗

arXiv2019

On ZpZp[u, v]-additive cyclic and constacyclic codes

Let $\mathbb{Z}_{p}$ be the ring of residue classes modulo a prime $p$. The $\mathbb{Z}_{p}\mathbb{Z}_{p}[u,v]$-additive cyclic codes of length $(\alpha,\beta)$ is identify as $\mathbb{Z}_{p}[u,v][x]$-submodule of $\mathbb{Z}_{p}[x]/\langle x^{\alpha}-1\rangle \times \mathbb{Z}_{p}[u,v][x]/\langle x^{\beta}-1\rangle$ where $\mathbb{Z}_{p}[u,v]=\mathbb{Z}_{p}+u\mathbb{Z}_{p}+v\mathbb{Z}_{p}$ with $u^{2}=v^{2}=uv=vu=0$. In this article, we obtain the complete sets of generator polynomials, minimal generating sets for cyclic codes with length $\beta$ over $\mathbb{Z}_{p}[u,v]$ and $\mathbb{Z}_{p}\mathbb{Z}_{p}[u,v]$-additive cyclic codes with length $(\alpha,\beta)$ respectively. We show that the Gray image of $\mathbb{Z}_{p}\mathbb{Z}_{p}[u,v]$-additive cyclic code with length $(\alpha,\beta)$ is either a QC code of length $4\alpha$ with index $4$ or a generalized QC code of length $(\alpha,3\beta)$ over $\mathbb{Z}_{p}$. Moreover, some structural properties like generating polynomials, minimal generating sets of $\mathbb{Z}_{p}\mathbb{Z}_{p}[u,v]$-additive constacyclic code with length $(\alpha,p-1)$ are determined.

cs.IT↗

arXiv2018

Skew cyclic codes over F_{p}+uF_{p}+\dots +u^{k-1}F_{p}

In this article, we study the skew cyclic codes over R_{k}=F_{p}+uF_{p}+\dots +u^{k-1}F_{p} of length n. We characterize the skew cyclic codes of length $n$ over R_{k} as free left R_{k}[x;θ]-submodules of R_{k}[x;θ]/\langle x^{n}-1\rangle and construct their generators and minimal generating sets. Also, an algorithm has been provided to encode and decode these skew cyclic codes.

cs.IT↗

arXiv2018

Skew cyclic and skew $(α_1 + uα_2 + vα_3 + uvα_4)$-constacyclic codes over $F_q + uF_q + vF_q + uvF_q$

In this note, we study skew cyclic and skew constacyclic codes over the ring $\mathcal{R}=F_{q}+uF_{q}+vF_{q}+uvF_{q}$ where $q=p^{m},$ $p$ is an odd prime, $u^{2}=u,~v^{2}=v,~uv=vu$. We show that Gray images of a skew cyclic and skew $α$-constacyclic code of length $n$ are skew quasi-cyclic code of length $4n$ over $F_{q}$ of index 4. Also, it is shown that skew $α$-constacyclic codes are either equivalent to $α$-constacyclic codes or $α$-quasi-twisted codes over $\mathcal{R}$. Further, structural properties, specially, generating polynomials and idempotent generators for skew cyclic and skew constacyclic codes are determined by decomposition method.

cs.IT↗

arXiv2017

Skew constacyclic codes over Fq+uFq+vFq

In this paper skew constacyclic codes over finite non-chain ring R = F_q+uF_q+vF_q, where q= p^m, p is an odd prime and u^{2}=u, v^{2}=v, uv = vu = 0 are studied. We show that Gray image of a skew alpha-constacyclic cyclic code of length n over R is a skew alpha-quasi-cyclic code of length $3n$ over F_{q} of index 3. It is also shown that skew alpha-constacyclic codes are either equivalent to alpha-constacyclic codes or alpha-quasi-twisted codes over R. Further, the structural properties of skew constacyclic over R are obtained by decomposition method.

cs.IT↗