The Maximum Length of Circuit Codes With Long Bit Runs and a New Characterization Theorem

TL;DR

This paper derives a formula for the maximum length of certain symmetric circuit codes with long bit runs, improves bounds for general circuit codes, and introduces a new characterization of spread-k circuit codes based on Deimer's theorem.

Contribution

It provides a new formula for maximum length of symmetric circuit codes with long bit runs and a novel characterization theorem for spread-k circuit codes.

Findings

Derived a maximum length formula for symmetric circuit codes with long bit runs.

Established an improved lower bound for general circuit codes.

Presented a new characterization theorem for spread-k circuit codes.

Abstract

We study circuit codes with long bit runs (sequences of distinct transitions) and derive a formula for the maximum length for an infinite class of symmetric circuit codes with long bit runs. This formula also results in an improved lower bound on the maximum length for an infinite class of circuit codes without restrictions on symmetry or bit run length. We also present a new characterization of circuit codes of spread $k$ based on a theorem of Deimer.