Cyclic Codes over the Integers Modulopm

✍ Scribed by Pramod Kanwar; Sergio R. López-Permouth

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 430 KB
• Volume: 3
• Category: Article
• ISSN: 1071-5797
• DOI: 10.1006/ffta.1997.0189

✦ Synopsis

The purpose of this paper is twofold. First, we generalize the results of Pless and Qian and those of Pless, Sole ´, and Qian for cyclic ‫ޚ‬ 4 -codes to cyclic ‫ޚ‬ p m -codes. Second, we establish connections between this new development and the results on cyclic ‫ޚ‬ p m -codes obtained by Calderbank and Sloane. We produce generators for the cyclic ‫ޚ‬ p m -codes which are analogs to those for cyclic ‫ޚ‬ 4 -codes. We show that these may be used to produce a single generator for such codes. In particular, this proves that the ring R n ϭ ‫ޚ‬ p m [x ]/(x n Ϫ 1) is principal, a result that had been previously announced with an incorrect proof. Generators for dual codes of cyclic ‫ޚ‬ p m -codes are produced from the generators of the corresponding cyclic ‫ޚ‬ p m -codes. In addition, we also obtain generators for the cyclic p m -ary codes induced from the idempotent generators for cyclic p -ary codes.