The Maximum Length and Isomorphism of Circuit Codes with Long Bit Runs

TL;DR

This paper extends the known formula for the maximum length of symmetric circuit codes with long bit runs to include even spread cases and proves that all such maximum codes are isomorphic within an infinite family.

Contribution

It generalizes Byrnes' formula to even spread and demonstrates isomorphism among all maximum length codes with long bit runs in an infinite family.

Findings

The maximum length formula applies to even spread cases.

All maximum length symmetric circuit codes with long bit runs are isomorphic in an infinite family.

The results extend previous work by Douglas and Byrnes.

Abstract

Recently, Byrnes presented a formula for the maximum length of a symmetric circuit code that has a long bit run and odd spread. Here we show that the formula is also valid when the spread is even. We also establish that all maximum length symmetric circuit codes with long bit runs are isomorphic for an infinite and nontrivial family of circuit codes, extending a previous result of Douglas.