On the Generation of Aperiodic and Periodic Necklaces via T-augmentation

Abstract

In a recent report, Gulliver and Speidel showed that complete sets of aperiodic and – in some cases periodic – necklaces can be generated for arbitrary lengths as fixed-length subsets of variable-length T-codes. The T-codes to which they applied this observation were specifically T-codes constructed by systematic T-augmentation, that is by T-augmentation sequences in which each T-expansion parameter is 1 (simple T-augmentation) and where the T-codes A _{(p1,p2,…,pi)} at each Taugmentation level i do not contain any codewords shorter than pi (strictly minimal T-augmentation). This present paper generalizes their result to arbitrary T-codes and formalizes their earlier result as a special case of the general result.