Binary and Non-Binary Self-Dual Sequences and Maximum Period Single-Track Gray Codes

TL;DR

This paper studies the structure and construction of binary and non-binary self-dual sequences, introduces new maximum period single-track Gray codes, and explores their applications and connections to feedback shift registers.

Contribution

It presents the first infinite families of maximum period non-binary single-track Gray codes of length p^t, expanding the known classes of such codes.

Findings

Constructed maximum period non-binary single-track Gray codes of length p^t

Analyzed recursive constructions for self-dual sequences

Connected these sequences to feedback shift register implementations

Abstract

Binary self-dual sequences have been considered and analyzed throughout the years, and they have been used for various applications. Motivated by a construction for single-track Gray codes, we examine the structure and recursive constructions for binary and non-binary self-dual sequences. The feedback shift registers that generate such sequences are discussed. The connections between these sequences and maximum period single-track codes are discussed. Maximum period non-binary single-track Gray codes of length $p^t$ and period $p^{p^t}$ are constructed. These are the first infinite families of maximum period codes presented in the literature.