arXiv:2606.22069

Hulls and sums of separable constacyclic codes over $\mathbb{F}_q \times (\mathbb{F}_q+v\mathbb{F}_q)$ and new quantum codes

Yu Qian, Yu Wang, Liqi Wang

Published Jun 20, 2026• 0 citations

Abstract

We establish the generator polynomials of the Euclidean and Hermitian duals of separable constacyclic codes over $\mathcal{S} = \mathbb{F}_q \times (\mathbb{F}_q+v\mathbb{F}_q)$, with $q$ an odd prime power and $v^2=v$, and we derive the generator polynomials of their Gray images, respectively. The generator polynomials of the Euclidean hulls and Hermitian hulls of separable constacyclic codes over $\mathcal{S}$ and their Gray images are presented, respectively. Furthermore, we provide the generator polynomials of the Euclidean sums and Hermitian sums of separable constacyclic codes and their Gray images, respectively. Finally, we propose two methods to yield quantum error-correcting codes (QECCs) from the hulls and sums of separable constacyclic codes over $\mathcal{S}$, and generate new QECCs that outperform the existing ones in terms of parameters.

DOI: 10.48550/arxiv.2606.22069

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arXiv:2606.22069

Hulls and sums of separable constacyclic codes over $\mathbb{F}_q \times (\mathbb{F}_q+v\mathbb{F}_q)$ and new quantum codesHulls and sums of separable constacyclic codes over $\mathbb{F}_q \times (\mathbb{F}_q+v\mathbb{F}_q)$ and new quantum codes

Yu Qian, Yu Wang, Liqi Wang

Published Jun 20, 2026• 0 citations

arXiv HTML PDF

Abstract

We establish the generator polynomials of the Euclidean and Hermitian duals of separable constacyclic codes over

\mathcal{S} = \mathbb{F}_q \times (\mathbb{F}_q+v\mathbb{F}_q)

, with

q

an odd prime power and

v^2=v

, and we derive the generator polynomials of their Gray images, respectively. The generator polynomials of the Euclidean hulls and Hermitian hulls of separable constacyclic codes over

\mathcal{S}

and their Gray images are presented, respectively. Furthermore, we provide the generator polynomials of the Euclidean sums and Hermitian sums of separable constacyclic codes and their Gray images, respectively. Finally, we propose two methods to yield quantum error-correcting codes (QECCs) from the hulls and sums of separable constacyclic codes over

\mathcal{S}

, and generate new QECCs that outperform the existing ones in terms of parameters.

DOI: 10.48550/arxiv.2606.22069

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EntropyApr 3, 20261 citation

Hulls and sums of separable constacyclic codes over

\mathbb{F}_q \times (\mathbb{F}_q+v\mathbb{F}_q)

and new quantum codes

We establish the generator polynomials of the Euclidean and Hermitian duals of separable constacyclic codes over

\mathcal{S} = \mathbb{F}_q \times (\mathbb{F}_q+v\mathbb{F}_q)

, with

q

an odd prime power and

v^2=v

, and we derive the generator polynomials of their Gray images, respectively. The generator polynomials of the Euclidean hulls and Hermitian hulls of separable constacyclic codes over

\mathcal{S}

\mathcal{S}

, and generate new QECCs that outperform the existing ones in terms of parameters.

Abstract In this paper, we study skew constacyclic codes over a class of non-chain rings

\mathcal{T}=\mathbb{F}_q[u_1,u_2,\ldots, u_r]/ \langle f_i(u_i), u_iu_j-u_ju_i\rangle_{i,j=1}^r

, where

q=p^m

p

is some odd prime,

m

is a positive integer and

f_i(u_i)

are non-constant polynomials which split into distinct linear factors. We discuss the structural properties of skew constacyclic codes over

\mathcal{T}

and their dual. We characterize (Euclidean and Hermitian) dual-containing skew constacyclic codes. These characterizations serve as a foundational framework for the development of techniques to construct quantum codes. Consequently, we derive plenty of new quantum codes including many Maximum Distance Separable (MDS) quantum codes, and many quantum codes with better parameters than existing ones. Our work further extends to the characterization of skew constacyclic Euclidean and Hermitian Linear Complementary Dual (LCD) codes over

\mathcal{T}

, and we establish that their Gray images also preserve the LCD property. From this analysis, we derive numerous Maximum Distance Separable (MDS) codes and Best-Known Linear Codes (BKLC) over

\mathbb{F}_q

. MSC Classification: MSC 11T06, MSC 81P70, MSC94B05, MSC 94B15, MSC 94B99

New non-binary quantum codes from skew constacyclic codes over the ring

\mathbb{F}_{p^m}+v\mathbb{F}_{p^m}+v^2 \mathbb{F}_{p^m}

In this article, we construct new non-binary quantum codes from skew constacyclic codes over finite commutative non-chain ring

\mathcal{R}= \mathbb{F}_{p^m}[v]/\langle v^3 =v \rangle

where

p

is an odd prime and

m \geq 1

. In order to obtain such quantum codes, first we study the structural properties of skew constacyclic codes and their Euclidean duals over the ring

\mathcal{R}

. Then a necessary and sufficient condition for skew constacyclic codes over

\mathcal{R}

to contain their Euclidean duals is established. Finally, with the help of CSS construction and using Gray map, many new non-binary quantum codes are obtained over

\mathbb{F}_{p^m}

QEC and EAQEC Codes from Hermitian Sums and Hulls of Cyclic Codes over

\mathbb{F}_2 \times (\mathbb{F}_2+v\mathbb{F}_2)

In this work, we determine the generator polynomials for the Hermitian hulls and Hermitian sums of cyclic codes defined over the composite ring

\mathbb{F}_2 \times (\mathbb{F}_2 + v\mathbb{F}_2)

, where

v^2 = v

. Based on these structures, we develop quantum error-correcting (QEC) codes by applying the Hermitian dual version of Quantum Construction~X to the obtained Hermitian hulls and sums. Moreover, by employing matrix product code methods on linear complementary dual (LCD) codes defined over the same ring, we derive families of entanglement-assisted quantum error-correcting (EAQEC) codes.

Quantum codes from constacyclic codes over

S_{k}

1\leq i,j\leq k

1 ≤ i , j ≤ k ,

q=p^{m}

q = p m , p is an odd prime. First, we define two new Gray maps

\phi _{k}

ϕ k and

\varphi _{k}

φ k , and study their Gray images. Further, we determine the structure of constacyclic codes and their dual codes, and give a necessary and sufficient conditions of constacyclic codes to contain their duals. Finally, we obtain some new quantum codes over

\mathbb{F}_{q}

F q by using CSS construction, and compare the constructed codes better than the existing literature.