Repeated-root cyclic codes

Abstract

In the theory of cyclic codes, it is common practice to require that (n,q)=1, where n is the word length and Fq is the alphabet. It is shown that the even weight subcodes of the shortened binary Hamming codes form a sequence of repeated-root cyclic codes that are optimal. In nearly all other cases, one does not find good cyclic codes by dropping the usual restriction that n and q must be relatively prime. This statement is based on an analysis for lengths up to 100. A theorem shows why this was to be expected, but it also leads to low-complexity decoding methods. This is an advantage, especially for the codes that are not much worse than corresponding codes of odd length. It is demonstrated that a binary cyclic code of length 2n (n odd) can be obtained from two cyclic codes of length n by the well-known | u|u+v| construction. This leads to an infinite sequence of optimal cyclic codes with distance 4. Furthermore, it is shown that low-complexity decoding methods can be used for these codes. The structure theorem generalizes to other characteristics and to other lengths. Some comparisons of the methods using earlier examples are given