Four Classes of Few-Weight Self-Orthogonal Codes and Their Applications for LCD Codes and Quantum Codes

Jul 16, 2026

Upload by Quinn

Self-orthogonal codes, few-weight codes, linear complementary dual (LCD) codes, and quantum codes are key objects in coding theory and cryptography. This manuscript introduces augmented codes built from defining sets and studies two such defining-set classes. It constructs one class of projective four-weight self-orthogonal codes and three classes of four-weight self-orthogonal codes, determines parameters of the dual codes, and derives applications including LCD codes and quantum codes with near-optimality and AMDS properties.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages19

Reading Time48 min

Introduction

Code fundamentals and optimality notions

Defining sets and augmented codes

Projective and self-orthogonal properties

Few-weight constructions and parameter analysis

Applications to LCD and quantum codes

arXiv :2607 .07 18 1v 1 [ cs .IT] 8 Jul 2026 1 Four classes of few-weight self-orthogonal codes and their applications for LCD codes and quantum codes Yue Huang, Zhonghao Liang, Chenlu Jia, Yongkang Wan and Qunying Liao Abstract Since self-orthogonal codes, few-weight codes, linear complementary dual codes(LCD codes, for short) and quantum codes have nice applications in coding theory and cryptography, they have received continuous attention. In 2024, by introducing the notion of the augment code, Heng et al.[30] constructed several classes of few-weight self-orthogonal codes basing on defining sets, which are introduced by Ding et al.[10] in 2007 . In this manuscript, for two classes of defining sets, we consider the corresponding augmented codes, construct a class of projective four-weight self-orthogonal codes and three classes of four-weight self-orthogonal codes. And for two classes of these four-weight self-orthogonal linear codes, we determine the parameters of their dual codes. As applications, we construct two classes of LCD codes and a class of quantum codes. In particular, we prove that there exists a class of these LCD codes whose dual codes are almost optimal LCD codes according to the sphere packing bound, and a class of quantum codes are AMDS according to the quantum Singleton bound. Index Terms Few-weight code; Weight distribution; Self-orthogonal code; LCD code; Quantum code. I. INTRODUCTION Let Fpm be the finite field with pm elements, where p is a prime and m is a positive integer. An [n, k, d] linear code C over Fpm is a k-dimensional linear subspace of Fnpm with minimum (Hamming) distance d. The weight wt(c) of a codeword c ∈ Cis the number of nonzero coordinates in c. Let Ai be the number of codewords c ∈ C with weight i. Then (1, A1 , ··· , An) is called the weight distribution of C and the polynomial 1 + A 1 z + ··· + Anzn is called the weight enumerator of C. If \#{i : Ai 0 , 1 ≤ i ≤ n} = t, then C is t-weight. If the Hamming weight of each codeword in C is divisible by p, then C isp-divisible. The dual code C ⊥ of C is defined by C ⊥ = 􀀈x ∈ Fnpm : x · c = 0 for all c ∈ C􀀉 . If the dual code of C has the minimal distance d⊥ ≥ 3 , then C is a projective code. If C ⊆ C ⊥ , then C is self-orthogonal; and if C ∩ C ⊥ = {0} , then C is alinear complementary dual code(LCD code, for short) . It is well-known that any linear code over Fpm has three fundamental parameters: the length n, the dimension k and the minimum Hamming distance d. A central goal in error-correcting code theory is to construct codes with good performance, i.e., simultaneously large n, k, d. Nevertheless, classical coding bounds show that these parameters cannot be maximized simultaneously, which introduces three separate definitions of optimality as follows: for any given [n, k, d]pm linear code C , if there is no [n − 1, k, d]pm linear code, then C is length-optimal; if there is no [n, k + 1, d]pm linear code, then C is dimension-optimal; if there is no [n, k, d + 1]pm linear code, then C is distance-optimal. In 2025, Chen et al.[50], defined the distance-almost optimal [n, k, d]pm linear code. Similarly, the dimension-almost optimal also can be defined, i.e., for any given [n, k, d]pm linear code C , if there exists an [n, k + 1, d]pm linear code, but there is no [n, k + 2, d]pm code, then C is dimension-almost optimal. So far, there are a lot of works on constructing optimal linear codes [24, 32, 47, 50] . Since self-orthogonal codes and few-weight codes play significant roles in coding theory and cryptography, they have been received considerable attention[3, 6, 11, 12, 14–16, 24, 29, 30, 32, 36, 47, 55–57] . Especially, in 2007, Ding et al. [9] introduced a new way to construct linear codes. Let D = {d1 , d2 , ··· , dn } ⊆ Fpm and defined CD = {(Tr(bd1 ) , Tr(bd2 ) , ··· , Tr(bdn )) : b ∈ Fpm } , where Tr is the trace function from Fpm to Fp. And D is called the defining set of CD . So far, there are many

FAQs

What kinds of codes does the manuscript focus on?

The manuscript studies self-orthogonal codes and few-weight codes, and uses them to construct linear complementary dual (LCD) codes and quantum codes.

How are augmented codes constructed in this work?

Augmented codes are defined from an original linear code by adding the all-ones vector scaled by elements of the base field, and the construction is applied to codes induced by defining sets.

What are the main application results for LCD codes and quantum codes?

The manuscript constructs classes of LCD codes whose dual codes are almost optimal under the sphere packing bound, and constructs quantum codes that achieve AMDS behavior according to the quantum Singleton bound.

Jul 16, 2026

Upload by Quinn