Conflict-Avoiding Codes of Prime Lengths and Cyclotomic Numbers

TL;DR

This paper explores the construction of conflict-avoiding codes of prime lengths using cyclotomic numbers, providing new results for certain prime lengths and advancing understanding in this area.

Contribution

It establishes a connection between conflict-avoiding codes of prime lengths and cyclotomic numbers, enabling the construction of optimal codes for new prime lengths.

Findings

Connected cyclotomic numbers with code construction methods

Constructed optimal conflict-avoiding codes for new prime lengths

Extended the known sizes of optimal codes for specific primes

Abstract

The problem to construct optimal conflict-avoiding codes of even lengths and the Hamming weight $3$ is completely settled. On the contrary, it is still open for odd lengths. It turns out that the prime lengths are the fundamental cases needed to be constructed. In the article, we study conflict-avoiding codes of prime lengths and give a connection with the so-called cyclotomic numbers. By having some nonzero cyclotomic numbers, a well-known algorithm for constructing optimal conflict-avoiding codes will work for certain prime lengths. As a consequence, we are able to answer the size of optimal conflict-avoiding code for a new class of prime lengths.