The structure of sidelobe-preserving operator groups

Abstract

This paper considers the structure of groups of operators preserving the aperiodic autocorrelation peak sidelobe level of the mth root of unity codes. These groups are shown to be helpful for efficient enumeration of codes by peak sidelobe level for a given m and given codeword length N. Another possible use is in narrowing the search space for the mth root of unity codes of a given length. In the binary case, it is shown that there is a single Abelian group of order 8 generated by sidelobe-preserving operators. Furthermore, it is shown that shared symmetry in the odd-length binary Barker codes can be discovered in a natural way by considering degeneracies of group actions. The group structure for m > 2 is shown to have higher complexity. Instead of a single group, there are m order-4m2 groups, and they are no longer Abelian. The structure of these groups is identified for any m > 2 and any positive length N.