On the algebraic structure of quasi-cyclic codes. II: Chain rings
This paper is Part II of an investigation of the algebraic structure of quasi-cyclic (QC) codes. QC codes over a finite commutative ring \(A\) are viewed as codes over the polynomial ring \[ R(A,m) := A[Y]/(Y^m-1). \] This latter ring can be decomposed into a direct sum of local rings. This can be achieved either by the Chinese Remainder Theorem or the Discrete Fourier Transform. The ring decomposition technique of Part I [IEEE Trans. Inf. Theory 47, 2751-2760 (2001; Zbl 1023.94015)] is extended to the case when the factors in the direct product decomposition are no longer fields but arbitrary chain rings. This includes not only the case of quasi-cyclic codes over rings but also the case of quasi-cyclic codes over fields whose co-index is no longer prime to the characteristic field. This approach allows for a systematic investigation of self-dual QC codes. In addition many well-known code constructions are obtained, as well as new ones (quinting and septing). Several examples are given, including the binary and ternary extended Golay codes. A new quaternary construction of the Leech lattice is derived.
scientific article; zbMATH DE number 1975931