WSEAS TRANSACTIONS ON MATHEMATICS

On Quantum Codes over Non Local Rings

Noureddine Essaidi, Abdelhamid Tadmori, Ossama El Abouti

Published Jan 2, 2024• 0 citations

Abstract

We study the structural properties of the ring R = F p + uF p + u 2 F p + u 3 F p , where p = 2 is a prime and u 4 = u 3 , as well as the linear codes over R. We investigate the generator's cyclic codes and their dual codes over R.An isometric Gray map from R to F 4 p is defined.We offer an equivalence condition that cyclic codes must satisfy over R in order to include their dual.Moreover, we establish the existence of quantum error-correcting codes based on cyclic codes over R. Finally, under various criteria such as cyclic codes length and generator polynomial degree, we build quantum error-correcting codes over R which their dimensions are divided by p and 2 4 .

DOI: 10.37394/23206.2024.23.3

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WSEAS TRANSACTIONS ON MATHEMATICS

On Quantum Codes over Non Local Rings

Noureddine Essaidi, Abdelhamid Tadmori, Ossama El Abouti

Published Jan 2, 2024• 0 citations

Publisher

Abstract

DOI: 10.37394/23206.2024.23.3

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In this paper, we extend the results given in [3] to a nonchain ring

R_p={\mathbb{F}}_p+v{\mathbb{F}}_p+{\cdots}+v^{p-1}{\mathbb{F}}_p

, where

v^p=v

and p is a prime. We determine the structure of the cyclic codes of arbitrary length over the ring

R_p

and study the structure of their duals. We classify cyclic codes containing their duals over

R_p

by giving necessary and sufficient conditions. Further, by taking advantage of the Gray map

{\pi}

defined in [4], we give the parameters of the quantum codes of length pn over

{\mathbb{F}}_p

which are obtained from cyclic codes over

R_p

. Finally, we illustrate the results by giving some examples.

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.