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

TL;DR

The paper investigates binary and non-binary self-dual sequences generated by complemented cycling registers and their role in constructing maximum-period single-track Gray codes. It develops operator-based methods to build and enumerate self-dual sequences, extends these constructions to non-binary alphabets, and presents recursive schemes to realize maximum-period STGCs for lengths $2^t$ and $p^t$ (with odd primes $p$). Key contributions include the $\mathbf{D}^{-2^n}$-driven expansion of SDS families, $m$-CCR$_n$ generalizations with explicit counting formulas, and seed-based recursive constructions for non-binary STGCs. The results lay groundwork for first infinite families of maximum-period STGCs and connect SDS structure to practical Gray-code design, with proofs deferred to a full version. These insights advance systematic design of high-length, high-period Gray codes across binary and non-binary domains.

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.

Figures (1)

• Figure 1: Feedback shift register of order $n$.

Theorems & Definitions (14)

• Theorem 1
• Lemma 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Example 1
• Theorem 6
• Theorem 7
• Lemma 2