Infinite families of 3-designs from special symmetric polynomials

被引：0

作者：

Xu, Guangkui ^{[1
,2
]}

Cao, Xiwang ^{[3
,4
]}

Luo, Gaojun ^{[3
,4
]}

Wu, Huawei ^{[5
,6
]}

机构：

[1] Anhui Jianzhu Univ, Sch Math & Phys, Hefei 230601, Peoples R China

[2] Anhui Jianzhu Univ, Operat Res & Data Sci Lab, Hefei, Peoples R China

[3] Nanjing Univ Aeronaut & Astronaut, Sch Math, Nanjing 210016, Peoples R China

[4] NUAA, MIIT, Key Lab Math Modeling & High Performance Comp Air, Nanjing 210016, Peoples R China

[5] Univ Paris VIII, Dept Math, F-93526 St Denis, France

[6] Univ Sorbonne Paris Nord, Lab Anal Geometry & Applicat LAGA, CNRS, UMR 7539, F-93430 Villetaneuse, France

来源：

DESIGNS CODES AND CRYPTOGRAPHY | 2024年

基金：

中国国家自然科学基金; 中国博士后科学基金;

关键词：

Symmetric polynomial; AMDS code; NMDS code; t-design; Projective general linear group;

D O I：

10.1007/s10623-024-01508-5

中图分类号：

TP301 [理论、方法];

学科分类号：

081202 ;

摘要：

Tang and Ding (IEEE Trans Inf Theory 67(1):244-254, 2021) opened a new direction of searching fort-designs from elementary symmetric polynomials, which are used to construct the first infinite family of linear codes supporting 4-designs. In this paper, we first study the properties of elementary symmetric polynomials with 6 or 7 variables over GF(3(m)). Based on them, we present more infinite families of 3-designs that contain some 3-designs with new parameters as checked by Magma for small numbers m. We also construct an infinite family of cyclic codes over GF(q(2))and prove that the codewords of any nonzero weight support a 3-design. In particular, we present an infinite family of 6-dimensional AMDS codes over GF(3(2m))holding an infinite family of 3-designs and an infinite family of 7-dimensionalNMDS codes over GF(3(2m))holding an infinite family of 3-designs.

引用

页码：4487 / 4509

页数：23

共 30 条

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