A New Method for Constructing Circuit Codes

TL;DR

This paper introduces a novel recursive method for constructing circuit codes with higher spread, achieving record lengths for certain parameters and improving bounds for others, with applications in error correction.

Contribution

It presents a new recursive construction technique for circuit codes that enhances code length and bounds, advancing the design of error-correcting codes.

Findings

Achieved record code lengths for spread 7 and 8 in dimensions 22 to 30.

Derived a new lower bound on circuit code length for spread 4 in high dimensions.

Improved existing bounds for circuit codes in large dimensions.

Abstract

Circuit codes are constructed from induced cycles in the graph of the $n$ dimensional hypercube. They are both theoretically and practically important, as circuit codes can be used as error correcting codes. When constructing circuit codes, the length of the cycle determines its accuracy and a parameter called the spread determines how many errors it can detect. We present a new method for constructing a circuit code of spread $k+1$ from a circuit code of spread $k$. This method leads to record code lengths for circuit codes of spread $k=7 \text{ and } 8$ in dimension $22\le n\le 30$. We also derive a new lower bound on the length of circuit codes of spread 4, improving upon the current bound for dimension $n\ge 86$.