Infinite Families of Cyclic and Negacyclic Codes Supporting 3-Designs

Interplay between coding theory and combinatorial <inline-formula> <tex-math notation="LaTeX">t </tex-math></inline-formula>-designs has been a hot topic for many years for combinatorialists and coding theorists. Some infinite families of cyclic codes supporting inf...

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Vydané v:IEEE transactions on information theory Ročník 69; číslo 4; s. 2341 - 2354
Hlavní autori: Wang, Xiaoqiang, Tang, Chunming, Ding, Cunsheng
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: New York IEEE 01.04.2023
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Predmet:
Codes
Coding
Coding theory
Combinatorial analysis
Constacyclic code
cyclic code
Generators
Hamming distances
Hamming weight
Indexes
Linear codes
Mathematics
negacyclic code
Steiner system
t-design
ISSN:0018-9448, 1557-9654
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Shrnutí:Interplay between coding theory and combinatorial <inline-formula> <tex-math notation="LaTeX">t </tex-math></inline-formula>-designs has been a hot topic for many years for combinatorialists and coding theorists. Some infinite families of cyclic codes supporting infinite families of 3-designs have been constructed in the past 50 years. However, no infinite family of negacyclic codes supporting an infinite family of 3-designs has been reported in the literature. This is the main motivation of this paper. Let <inline-formula> <tex-math notation="LaTeX">q=p^{m} </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">p </tex-math></inline-formula> is an odd prime and <inline-formula> <tex-math notation="LaTeX">m \geq 2 </tex-math></inline-formula> is an integer. The objective of this paper is to present an infinite family of cyclic codes over <inline-formula> <tex-math notation="LaTeX">{\mathrm {GF}}(q) </tex-math></inline-formula> supporting an infinite family of 3-designs and two infinite families of negacyclic codes over <inline-formula> <tex-math notation="LaTeX">{\mathrm {GF}}(q^{2}) </tex-math></inline-formula> supporting two infinite families of 3-designs. The parameters and the weight distributions of these codes are determined. The subfield subcodes of these negacyclic codes over <inline-formula> <tex-math notation="LaTeX">{\mathrm {GF}}(q) </tex-math></inline-formula> are studied. Three infinite families of almost MDS codes are also presented. A constacyclic code over <inline-formula> <tex-math notation="LaTeX">{\mathrm {GF}}(4) </tex-math></inline-formula> supporting a 4-design and seven open problems are also presented in this paper.
Bibliografia:ObjectType-Article-1
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ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2022.3222474