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
$n$
cyclic shifts of every given codeword are all distinct, where
$n$
is the code length. Optical orthogonal codes (OOCs) are constructed from constant weight binary HCCs by picking exactly one member from the
$n$
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’s inequality on the concentration of Lipschitz functions and the independence number of locally sparse graphs.