Affiliations: Department of Electrical and Computer Engineering,
University of Victoria, Victoria, BC, Canada. E-mail: a.gulliver@ieee.org | School of Mathematics, University College Dublin,
Belfield, Dublin 4, Ireland. E-mail: Isaiah.Makwakwa@ucdconnect.ie | Department of Computer Science, The University of
Auckland, Auckland, New Zealand. E-mail: ulrich@cs.auckland.ac.nz
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.
