Improved Gilbert-Varshamov bounds for hopping cyclic codes and optical orthogonal codes

TL;DR

This paper presents improved lower bounds for hopping cyclic codes and optical orthogonal codes, enhancing existing bounds by linear and quadratic factors respectively, with implications for related code families.

Contribution

It introduces new Gilbert-Varshamov bounds for HCCs and OOCs, significantly improving previous bounds using probabilistic and graph-theoretic methods.

Findings

Improved lower bounds for HCCs by a linear factor of code length.

Enhanced bounds for OOCs by a quadratic factor of code length.

Extended bounds to frequency hopping sequences and uncorrelated codes.

Abstract

Hopping cyclic codes (HCCs) are (non-linear) cyclic codes with the additional property that the $n$ cyclic shifts of every given codeword are all distinct, where $n$ is the code length. Constant weight binary hopping cyclic codes are also known as optical orthogonal codes (OOCs). HCCs and OOCs have various practical applications and have been studied extensively over the years. The main concern of this paper is to present improved Gilbert-Varshamov type lower bounds for these codes, when the minimum distance is bounded below by a linear factor of the code length. For HCCs, we improve the previously best known lower bound of Niu, Xing, and Yuan by a linear factor of the code length. For OOCs, we improve the previously best known lower bound of Chung, Salehi, and Wei, and Yang and Fuja by a quadratic factor of the code length. As by-products, we also provide improved lower bounds for frequency hopping sequences sets and error-correcting weakly mutually uncorrelated codes. Our proofs are based on tools from probability theory and graph theory, in particular the McDiarmid's inequality on the concentration of Lipschitz functions and the independence number of locally sparse graphs.