Assume that $q$ is a prime power and $m\geq 2$ is a positive integer. Cyclic codes over $\mathbb{F}_{q^{2m}}$ of length $n=\frac{q^{2m}-1}{\rho }$ with $\rho\mid (q-1)$ , and constacyclic codes over $\mathbb{F}_{q^{2m}}$ of length $n=\frac{q^{2m}-1}{\rho }$ with $\rho\mid (q+1)$ are considered in this paper, respectively. Two classes of quantum codes are derived from the images of these codes by the Hermitian construction. Compared with the previously known quantum codes, the quantum codes in our scheme have better parameters.Assume that $q$ is a prime power and $m\geq 2$ is a positive integer. Cyclic codes over $\mathbb{F}_{q^{2m}}$ of length $n=\frac{q^{2m}-1}{\rho }$ with $\rho\mid (q-1)$, and constacyclic codes over $\mathbb{F}_{q^{2m}}$ of length $n=\frac{q^{2m}-1}{\rho }$ with $\rho\mid (q+1)$ are considered in this paper, respectively. Two classes of quantum codes are derived from the images of these codes by the Hermitian construction. Compared with the previously known quantum codes, the quantum codes in our scheme have better parameters.

Quantum Information and ComputationJan 1, 20231 citation

Computable Derivative-Twisted Intersection Dimensions of Repeated-Root Cyclic Codes via Lucas-Refined Truncation

Ampol Duangpan, Ratinan Boonklurb, Phiraphat Sutthimat

The Hasse derivative image of a repeated-root cyclic code is typically non-cyclic. This prevents a direct ideal-theoretic calculation of derivative-twisted intersections and first leads to cyclic containers and two-sided bounds. This paper shows that, after expanding codewords in Hasse–Taylor coordinates at the roots of the underlying polynomial, the higher-order Hasse derivative operator decomposes into independent local blocks and its image becomes a coordinate subspace. This yields exact closed-form dimensions, expressed entirely in terms of the generator multiplicities and the base-prime digits of the derivative order, for the derivative image, for the derivative-twisted intersection of two codes and its self-intersection specialization, and for the smallest cyclic code containing the image together with its non-cyclic defect. As a consequence, for every positive derivative order in the admissible range, the derivative image is cyclic only when it is zero, and the multiplicity-drop and Lucas-refined containers developed here, as well as the rank of the derivative operator, are recovered as immediate relaxations or special cases. Applying the intersection-pair construction to a code paired with its derivative image produces entanglement-assisted quantum error-correcting codes whose dimension and entanglement cost are exact functions of the multiplicity digits and the derivative order, so that the derivative order tunes the entanglement consumption. For the minimum distance, a punctured matrix-product decomposition gives an exact formula for the derivative image in terms of the minimum distances of a finite family of simple-root constituents of length n0. Combining this formula with the corresponding constituent formula for the dual gives the complete distance parameter of the resulting EAQECCs without enumerating the non-cyclic derivative images.The Hasse derivative image of a repeated-root cyclic code is typically non-cyclic. This prevents a direct ideal-theoretic calculation of derivative-twisted intersections and first leads to cyclic containers and two-sided bounds. This paper shows that, after expanding codewords in Hasse–Taylor coordinates at the roots of the underlying polynomial, the higher-order Hasse derivative operator decomposes into independent local blocks and its image becomes a coordinate subspace. This yields exact closed-form dimensions, expressed entirely in terms of the generator multiplicities and the base-prime digits of the derivative order, for the derivative image, for the derivative-twisted intersection of two codes and its self-intersection specialization, and for the smallest cyclic code containing the image together with its non-cyclic defect. As a consequence, for every positive derivative order in the admissible range, the derivative image is cyclic only when it is zero, and the multiplicity-drop and Lucas-refined containers developed here, as well as the rank of the derivative operator, are recovered as immediate relaxations or special cases. Applying the intersection-pair construction to a code paired with its derivative image produces entanglement-assisted quantum error-correcting codes whose dimension and entanglement cost are exact functions of the multiplicity digits and the derivative order, so that the derivative order tunes the entanglement consumption. For the minimum distance, a punctured matrix-product decomposition gives an exact formula for the derivative image in terms of the minimum distances of a finite family of simple-root constituents of length n0. Combining this formula with the corresponding constituent formula for the dual gives the complete distance parameter of the resulting EAQECCs without enumerating the non-cyclic derivative images.

MathematicsAug 7, 20260 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

Assume that

q

is a prime power and

m\geq 2

is a positive integer. Cyclic codes over

\mathbb{F}_{q^{2m}}

of length

n=\frac{q^{2m}-1}{\rho }

with

\rho\mid (q-1)

, and constacyclic codes over

\mathbb{F}_{q^{2m}}

of length

n=\frac{q^{2m}-1}{\rho }

with

\rho\mid (q+1)

are considered in this paper, respectively. Two classes of quantum codes are derived from the images of these codes by the Hermitian construction. Compared with the previously known quantum codes, the quantum codes in our scheme have better parameters.

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.