Quasi-Cyclic Codes Over Finite Chain Rings
Jian Gao, Linzhi Shen, Fang-Wei Fu
TL;DR
This paper explores the structure and properties of quasi-cyclic codes over finite chain rings, providing bounds on their minimum distance and methods to enumerate and generate 1-generator codes.
Contribution
It introduces new structural insights, trace representations, and enumeration techniques for 1-generator quasi-cyclic codes over finite chain rings.
Findings
Lower bounds on minimum Hamming distance of QC codes
Structural properties of 1-generator QC codes
Enumeration and unique generator determination for 1-generator QC codes
Abstract
In this paper, we mainly consider quasi-cyclic (QC) codes over finite chain rings. We study module structures and trace representations of QC codes, which lead to some lower bounds on the minimum Hamming distance of QC codes. Moreover, we investigate the structural properties of 1-generator QC codes. Under some conditions, we discuss the enumeration of 1-generator QC codes and describe how to obtain the one and only one generator for each 1-generator QC code.
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Taxonomy
TopicsCoding theory and cryptography · graph theory and CDMA systems · Finite Group Theory Research