Self-orthogonal codes have attracted considerable attention owing to their applications in quantum error-correcting codes, linear complementary dual codes, and a variety of other fields. In this paper, we construct new families of self-orthogonal codes and self-orthogonal minimal codes by establishing criteria that characterize the self-orthogonality of certain linear codes. To this end, we first establish several new criteria for linear codes arising from the defining-set construction to be self-orthogonal. More specifically, for $q > 3$ , we show that the code $\mathcal{C}_D$ is self-orthogonal whenever the defining set $D$ is $G$ -invariant, where $G \subseteq \mathbb{F}_q^*$ and $|G| > 2$ . For $q = 2, 3$ , we characterize the self-orthogonality of $\mathcal{C}_D$ using certain character sums. By combining these criteria with partial difference sets, we construct several new families of self-orthogonal codes, which lead to optimal or almost optimal quantum codes. Secondly, via the action of a multiplicative subgroup $G \subseteq \mathbb{F}_q^*$ with $|G| > 2$ on the columns of a projective linear code, we construct self-orthogonal codes with flexible parameters. From their augmented codes, we derive quantum codes. By choosing suitable projective codes, we obtain optimal quantum codes with high parametric flexibility. Thirdly, we employ the characteristic function method to construct linear codes and establish criterion for their self-orthogonality. Based on this, we construct several classes of self-orthogonal minimal codes that violate the Ashikhmin-Barg condition, using vectorial dual-bent functions and $s$ -plateaued functions.Self-orthogonal codes have attracted considerable attention owing to their applications in quantum error-correcting codes, linear complementary dual codes, and a variety of other fields. In this paper, we construct new families of self-orthogonal codes and self-orthogonal minimal codes by establishing criteria that characterize the self-orthogonality of certain linear codes. To this end, we first establish several new criteria for linear codes arising from the defining-set construction to be self-orthogonal. More specifically, for $q > 3$, we show that the code $\mathcal{C}_D$ is self-orthogonal whenever the defining set $D$ is $G$-invariant, where $G \subseteq \mathbb{F}_q^*$ and $|G| > 2$. For $q = 2, 3$, we characterize the self-orthogonality of $\mathcal{C}_D$ using certain character sums. By combining these criteria with partial difference sets, we construct several new families of self-orthogonal codes, which lead to optimal or almost optimal quantum codes. Secondly, via the action of a multiplicative subgroup $G \subseteq \mathbb{F}_q^*$ with $|G| > 2$ on the columns of a projective linear code, we construct self-orthogonal codes with flexible parameters. From their augmented codes, we derive quantum codes. By choosing suitable projective codes, we obtain optimal quantum codes with high parametric flexibility. Thirdly, we employ the characteristic function method to construct linear codes and establish criterion for their self-orthogonality. Based on this, we construct several classes of self-orthogonal minimal codes that violate the Ashikhmin-Barg condition, using vectorial dual-bent functions and $s$-plateaued functions.

arXiv:2607.20805Jul 23, 20260 citations

A new class of linear codes with two-level structure

Haiyan Wang

Journal of Electronics (China)Jan 1, 20000 citations

Construction of Hermitian Self-Orthogonal Codes and Application

Yuezhen Ren, Ruihu Li, Hao Song

We introduce some methods for constructing quaternary Hermitian self-orthogonal (HSO) codes, and construct quaternary [n, 5] HSO for 342≤n≤492. Furthermore, we present methods of constructing Hermitian linear complementary dual (HLCD) codes from known HSO codes, and obtain many HLCD codes with good parameters. As an application, 31 classes of entanglement-assisted quantum error correction codes (EAQECCs) with maximal entanglement can be obtained from these HLCD codes. These new EAQECCs have better parameters than those in the literature.We introduce some methods for constructing quaternary Hermitian self-orthogonal (HSO) codes, and construct quaternary [n, 5] HSO for 342≤n≤492. Furthermore, we present methods of constructing Hermitian linear complementary dual (HLCD) codes from known HSO codes, and obtain many HLCD codes with good parameters. As an application, 31 classes of entanglement-assisted quantum error correction codes (EAQECCs) with maximal entanglement can be obtained from these HLCD codes. These new EAQECCs have better parameters than those in the literature.

MathematicsJul 5, 20242 citations

Three families of self-orthogonal codes and their application in optimal quantum codes

Dexiang Li, Ziling Heng, Chengju Li

Discrete MathematicsAug 3, 202312 citations

q > 3

, we show that the code

\mathcal{C}_D

is self-orthogonal whenever the defining set

D

G

-invariant, where

G \subseteq \mathbb{F}_q^*

and

|G| > 2

. For

q = 2, 3

, we characterize the self-orthogonality of

\mathcal{C}_D

using certain character sums. By combining these criteria with partial difference sets, we construct several new families of self-orthogonal codes, which lead to optimal or almost optimal quantum codes. Secondly, via the action of a multiplicative subgroup

G \subseteq \mathbb{F}_q^*

with

|G| > 2

on the columns of a projective linear code, we construct self-orthogonal codes with flexible parameters. From their augmented codes, we derive quantum codes. By choosing suitable projective codes, we obtain optimal quantum codes with high parametric flexibility. Thirdly, we employ the characteristic function method to construct linear codes and establish criterion for their self-orthogonality. Based on this, we construct several classes of self-orthogonal minimal codes that violate the Ashikhmin-Barg condition, using vectorial dual-bent functions and

s

-plateaued functions.