In this paper, we investigate the Gray images of codes over chain rings, leading to the derivation of infinite families of self-orthogonal linear codes over the residue field $\mathbb{F}_q$ . We determine the parameters of optimal self-orthogonal and divisible linear codes. Additionally, we study the Gray images of quasi-twisted codes, resulting in some self-orthogonal Griesmer quasi-cyclic codes. Finally, we employ the CSS construction to derive some quantum codes based on self-orthogonal linear codes. Received: 19 September 2023 | Accepted: 18 January 2024In this paper, we investigate the Gray images of codes over chain rings, leading to the derivation of infinite families of self-orthogonal linear codes over the residue field $\mathbb{F}_q$. We determine the parameters of optimal self-orthogonal and divisible linear codes. Additionally, we study the Gray images of quasi-twisted codes, resulting in some self-orthogonal Griesmer quasi-cyclic codes. Finally, we employ the CSS construction to derive some quantum codes based on self-orthogonal linear codes. Received: 19 September 2023 | Accepted: 18 January 2024

Journal of Algebra Combinatorics Discrete Structures and ApplicationsMay 6, 20240 citations

Encoding and Construction of Quantum Codes from $(γ,Δ)$ -cyclic Codes over a Class of Non-chain RingsEncoding and Construction of Quantum Codes from $(γ,Δ)$-cyclic Codes over a Class of Non-chain Rings

Om Prakash, Shikha Patel, Habibul Islam

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 $(γ,Δ)$ -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 $γ$ is an automorphism of $\mathscr{R}_{q,s}$ , $Δ$ is a $γ$ -derivation of $\mathscr{R}_{q,s}$ and $1\leq i\neq j\leq s$ for a positive integer $s$ . Here, we show that a $(γ,Δ)$ -cyclic code of length $n$ over $\mathscr{R}_{q,s}$ is the direct sum of $(θ,\Im)$ -cyclic codes of length $n$ over $\mathbb{F}_q$ , where $θ$ is an automorphism of $\mathbb{F}_q$ and $\Im$ is a $θ$ -derivation of $\mathbb{F}_q$ . Further, necessary and sufficient conditions for both $(γ,Δ)$ -cyclic and $(θ,\Im)$ -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.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 $(γ,Δ)$-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 $γ$ is an automorphism of $\mathscr{R}_{q,s}$, $Δ$ is a $γ$-derivation of $\mathscr{R}_{q,s}$ and $1\leq i\neq j\leq s$ for a positive integer $s$. Here, we show that a $(γ,Δ)$-cyclic code of length $n$ over $\mathscr{R}_{q,s}$ is the direct sum of $(θ,\Im)$-cyclic codes of length $n$ over $\mathbb{F}_q$, where $θ$ is an automorphism of $\mathbb{F}_q$ and $\Im$ is a $θ$-derivation of $\mathbb{F}_q$. Further, necessary and sufficient conditions for both $(γ,Δ)$-cyclic and $(θ,\Im)$-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.

arXiv:2404.01904Apr 2, 20240 citations

A new construction of quantum codes from quasi-cyclic codes over finite fields

Soumak Biswas, Maheshanand Bhaintwal

Indian Journal of Pure and Applied MathematicsMay 16, 20223 citations

In this paper, we investigate the Gray images of codes over chain rings, leading to the derivation of infinite families of self-orthogonal linear codes over the residue field

\mathbb{F}_q

. We determine the parameters of optimal self-orthogonal and divisible linear codes. Additionally, we study the Gray images of quasi-twisted codes, resulting in some self-orthogonal Griesmer quasi-cyclic codes. Finally, we employ the CSS construction to derive some quantum codes based on self-orthogonal linear codes. Received: 19 September 2023 | Accepted: 18 January 2024

Encoding and Construction of Quantum Codes from

(γ,Δ)

-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

(γ,Δ)

-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

γ

is an automorphism of

\mathscr{R}_{q,s}

Δ

is a

γ

-derivation of

\mathscr{R}_{q,s}

and

1\leq i\neq j\leq s

for a positive integer

s

. Here, we show that a

(γ,Δ)

-cyclic code of length

n

over

\mathscr{R}_{q,s}

is the direct sum of

(θ,\Im)

-cyclic codes of length

n

over

\mathbb{F}_q

, where

θ

is an automorphism of

\mathbb{F}_q

and

\Im

is a

θ

-derivation of

\mathbb{F}_q

. Further, necessary and sufficient conditions for both

(γ,Δ)

-cyclic and

(θ,\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.