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

Hopping cyclic codes (HCCs) are (non-linear) cyclic codes with the additional property that the <italic>n</italic> cyclic shifts of every given codeword are all distinct, where <italic>n</italic> is the code length. Optical orthogonal codes (OOCs) are constructed from constant weight binary HCCs by picking exactly one member from the <italic>n</italic> cyclic shifts of every codeword. HCCs and OOCs have various practical applications and have been studied extensively over the years. In this paper, we present improved Gilbert-Varshamov type lower bounds on the size of both 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 multiplicative 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 also by a multiplicative linear factor of the code length. Our proofs are based on tools from probability theory and graph theory, in particular the McDiarmid&#x2019;s inequality on the concentration of Lipschitz functions and the independence number of locally sparse graphs.