Bounds and Constructions for Optimal $(n, \{3, 4, 5\}, \Lambda _a, 1, Q)$ -OOCs

Abstract

Let $W=\{w_{1}, \ldots , w_{r}\}$ be a set of positive integers, $\lambda _{c}$ a positive integer, $\Lambda _{a}=(\lambda _{a}^{(1)}, \ldots \lambda _{a}^{(r)})$ an $r$ -tuple of positive integers, and $Q=(q_{1}, \ldots q_{r})$ an $r$ -tuple of positive rational numbers whose sum is 1. In 1996, Yang introduced variable-weight optical orthogonal code, $(n, W, \Lambda _{a}, \lambda _{c}, Q)$ -OOC, for multimedia optical CDMA systems with multiple quality of service (QoS) requirements. Some work had been done on the constructions of optimal $(n, W, \Lambda _{a}, 1, Q)$ -OOCs with unequal auto-correlation constraints for $W=\{3, 4\}$ and {3, 5}, while little is known on optimal $(n, W, \Lambda _{a}, 1, Q)$ -OOCs for $|W|\geq 3$ . In this paper, we focus our main attentions on $(n, \{3, 4, 5\}, \Lambda _{a}, 1, Q)$ -OOCs with $\Lambda _{a}\in \{(2, 1, 1), (2, 1, 2), (2, 2, 1), (2, 2, 2)\}$ . Tight upper bounds on the maximum code size of $(n, \{3, 4, 5\}, \Lambda _{a}, 1, Q)$ -OOCs are obtained, and infinite classes of optimal $(n, \{3, 4, 5\}, \Lambda _{a}, 1, Q)$ -OOCs are constructed.