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.
翻译:Hopping 自行车代码(HCC)是(非线性)自行车代码,其附加属性是,每个特定编码的最小距离由代码长度的线性系数约束在下方,每个编码的周期值均为不同的,其值为美元。常量重量双车自行车代码也称为光正方码(OOCs),常量二进制自行车代码(OOCs)也称为光正方码(OOCs),HCCs和OOCs具有各种实际应用性,多年来经过广泛研究。本文的主要关切是提出改进的Gilbert-Varshamov型代码的下限,当每个编码的最小距离由代码长度的线性系数限制在下方时。对于 HCCs,我们改进了以前已知的Niu、Xing和 ⁇ 的下限,以代码长度线为线性;对于OOCs,我们改进了以前已知的最低的Chung、Salehi和We、Yang和Fuja的下限,以代码的四重系数因素为基础。作为副产品,我们还改进了频率定序的下线性下限,错误校正正正数根据当地的模型和正数的模型和正数标准的模型,我们的证据依据依据了M正正正数的数值。