A class of 1-generator quasi-cyclic codes over finite chain rings

Abstract

Using another method, which is different with the one given by Cao [1-generator quasi-cyclic (QC) codes over finite chain rings, Appl. Algebra Eng. Commun. Comput. 24 (2013), pp. 53–72], we investigate the structural properties of a class of 1-generator QC codes over finite chain rings. We give the structure of the annihilator of 1-generator QC codes, the conditions for 1-generator QC codes to be free and the minimum distance bounds on 1-generator QC codes. Under some conditions, we also 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.

Keywords:

• finite chain rings
• 1-generator quasi-cyclic codes
• minimum distance
• enumeration
• generator

2000 AMS Subject Classifications:

• 11T71
• 94B05
• 94B15

Acknowledgments

The first author will thank Professor Cao for his useful suggestions on Section 4 in this paper. The authors would like to thank the two anonymous referees and the editor for their valuable comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Funding

This research is supported by the National Key Basic Research Program of China [Grant No. 2013CB834204]; National Natural Science Foundation of China [Grant No. 61171082].