Defence Science Journal

New Quantum and LCD Codes over Finite Fields of Even Characteristic

Habibul Islam, Om Prakash

Published Sep 2, 2021• 15 citations

Abstract

For an integer m ≥ 1, we study cyclic codes of length l over a commutative non-chain ring F2m + uF2m , where u2 = u . With a new Gray map and Euclidean dual-containing cyclic codes, we provide many new and superior codes to the best-known quantum error-correcting codes. Also, we characterise LCD codes of length l with respect to their generator polynomials and prove that F2m − image of an LCD code of length l is an LCD code of length 2l . Finally, we provide several optimal LCD codes from the Gray images of LCD codes over F2m + uF2m .

DOI: 10.14429/dsj.71.16641

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Defence Science Journal

New Quantum and LCD Codes over Finite Fields of Even Characteristic

Habibul Islam, Om Prakash

Published Sep 2, 2021• 15 citations

Publisher

Abstract

DOI: 10.14429/dsj.71.16641

Related papers

On the Structure of Cyclic Codes Over $\mathbb{F}_{q}{RS}$ and Applications in Quantum and LCD Codes ConstructionsOn the Structure of Cyclic Codes Over $\mathbb{F}_{q}{RS}$ and Applications in Quantum and LCD Codes Constructions

Hai Quang Dinh, Tushar Bag, Ashish Kumar Upadhyay +2 more

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Akanksha, Anuj Kumar Bhagat, Ritumoni Sarma

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arXiv:2501.01708Jan 3, 20250 citations

On the Structure of Cyclic Codes Over

\mathbb{F}_{q}{RS}

and Applications in Quantum and LCD Codes Constructions

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.

Let

\mathcal{T} = {\mathbb{F}_{{p^s}}} + {u_1}{\mathbb{F}_{{p^s}}} + {u_2}{\mathbb{F}_{{p^s}}} + {u_3}{\mathbb{F}_{{p^s}}} + u_1^2{\mathbb{F}_{{p^s}}}

be a finite non-chain ring. In this paper, we propose a method of constructing quantum codes and LCD codes from constacyclic codes over

\mathcal{T}

. The first codes are obtained via the Gray map and the Calderbank-Shor-Steane construction from Euclidean dualcontaining constacyclic codes over

\mathcal{T}

. The second codes are obtained as the gray image of consta cyclic codes of LCD codes over

\mathcal{T}

(Θ, Δ_Θ, \mathbf{a})

-cyclic codes over

\mathbb{F}_q^l

and their applications in the construction of quantum codes

In this article, for a finite field

\mathbb{F}_q

and a natural number

l,

let

\mathcal{R}

denote the product ring

\mathbb{F}_q^l.

Firstly, for an automorphism

Θ

\mathcal{R},

Θ

-derivation

Δ_Θ

\mathcal{R}

and for a unit

\mathbf{a}

\mathcal{R},

we study

(Θ, Δ_Θ, \mathbf{a})

-cyclic codes over

\mathcal{R}.

In this direction, we give an algebraic characterization of a

(Θ, Δ_Θ, \mathbf{a})

-cyclic code over

\mathcal{R}

, determine its generator polynomial, and find its decomposition over

\mathbb{F}_q.

Secondly, we give a necessary and sufficient condition for a

(Θ, 0, \mathbf{a})

-cyclic code to be Euclidean dual-containing code over

\mathcal{R}.

Thirdly, we study Gray maps and obtain several MDS and optimal linear codes over

\mathbb{F}_q

as Gray images of

(Θ, Δ_Θ, \mathbf{a})

-cyclic codes over

\mathcal{R}.

Moreover, we determine orthogonality preserving Gray maps and construct Euclidean dual-containing codes with good parameters. Lastly, as an application, we construct MDS and almost MDS quantum codes by employing the Euclidean dual-containing and annihilator dual-containing CSS constructions.