Asian-European Journal of Mathematics

New quantum codes from cyclic codes over finite chain ring of length 3

Brahim Boudine, Jamal Laaouine, M. E. Charkani

Published Dec 23, 2020• 1 citation

Abstract

Let [Formula: see text] be a commutative finite chain ring with [Formula: see text] is its maximal ideal, [Formula: see text] is its residual field and [Formula: see text] the index of nilpotency of [Formula: see text]. In this paper, we compute the dual code of some types of cyclic codes over [Formula: see text] when [Formula: see text]. Afterward, we show how extract some new quantum codes from these cyclic codes.

DOI: 10.1142/s1793557121501758

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Asian-European Journal of Mathematics

New quantum codes from cyclic codes over finite chain ring of length 3

Brahim Boudine, Jamal Laaouine, M. E. Charkani

Published Dec 23, 2020• 1 citation

Publisher

Abstract

DOI: 10.1142/s1793557121501758

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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.

For a prime p and a positive integer e, let

\mathbb {F}_{p^{e}}

be the finite field of order

p^{e}

and$\mathfrak {R}_{\ell }

be a non-chain ring given by

\mathfrak {R}_{\ell } :=\mathbb {F}_{p^{e}}[v]/\langle v^{\ell } -1\rangle,~\ell \gt 1

. This study presents the

(\theta _{1},\delta _{1},\beta _{1})

-cyclic codes over

\mathfrak {R}_{\ell }

and

(\theta _{i},\delta _{i},\beta _{i})

-cyclic codes over

\mathbb {F}_{p^{e}}\mathfrak {R}_{\ell }

for

i=0,1

. Further, we study the application of these codes in the construction of quantum codes. Towards this, we define a Gray map to find images of the elements of

\mathfrak {R}_{\ell }

and

\mathbb {F}_{p^{e}}\mathfrak {R}_{\ell }

over the copies of

\mathbb {F}_{p^{e}}$, and then establish the dual-containing conditions for these codes. Finally, using the CSS construction and a propagation rule, we find many non-binary quantum and classical codes with better parameters.