These groups are shown to be helpful for efficient enumeration of codes by peak sidelobe level for a given m and given code length N. This chapter considers the structure of groups of operators preserving the aperiodic autocorrelation peak sidelobe level of mth-root codes. The group structure for m even and N even is left unresolved. When m is even, the structure for odd code lengths N is identified. All m groups are identified for any odd m. In particular, it is shown that there are 4mĢ groups. Moving to general mth-roots codes, it is shown that results found for the quad-phase case generalize quite well. Group structure is identified for the cases of odd code lengths N, leaving group structure for even-length cases mostly unresolved. The group structure for m = 4 (the quad-phase case) is shown to have higher complexity in fact, instead of a single group, there are four groups (two pairs of isomorphic groups), and they are no longer Abelian. Furthermore, it is shown that shared symmetry in the binary Barker codes can be discovered in a natural way by considering degeneracies of group actions. In the binary case, it is shown that there is a single Abelian group of order 8 generated by sidelobe-preserving operators.
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