Symmetry Implies Isomorphism for Certain Maximum Length Circuit Codes

TL;DR

This paper extends a classic theorem on maximum length circuit codes, showing that symmetry implies isomorphism in certain cases, thereby broadening understanding of code structure for specific parameters.

Contribution

It generalizes Douglas's theorem to even spread values, proving all maximum length symmetric circuit codes are isomorphic under new conditions.

Findings

All maximum length symmetric circuit codes are isomorphic for even spread k ≥ 4 and specific dimensions.

Extension of Douglas's theorem to new parameter ranges.

Provides a unified understanding of code isomorphism under symmetry constraints.

Abstract

A classic result due to Douglas establishes that, for odd spread $k$ and dimension $d=\frac{1}{2}(3k+3)$, all maximum length $(d,k)$ circuit codes are isomorphic. Using a recent result of Byrnes we extend Douglas's theorem to prove that, for $k$ even $\ge 4$ and $d=\frac{1}{2}(3k+4)$, all maximum length symmetric $(d,k)$ circuit codes are isomorphic.