Assigning codes in wireless networks: bounds and scaling properties

In the Code Division Multiple Access (CDMA) framework, collisions that can occur in wireless networks are eliminated by assigning orthogonal codes to stations, a problem equivalent to that of coloring graphs associated to the physical network. In this paper we present new upper and lower bounds for two versions of the problem (hidden and primary collision avoidance - HP-CA - or hidden collision avoidance only - H-CA). In particular, optimal assignments for special topologies and heuristics for general topologies are proposed. The schemes show better average results with respect to existing alternatives. Furthermore, the gaps between the upper bound given by the heuristic solution, the lower bound obtained from the maximum-clique problem, and the optimal solution obtained by branch and bound are investigated in the different settings. A scaling law is then proposed to explain the relations between the number of codes needed in Euclidean networks with different station densities and connection distances. The substantial difference between the two versions HP-CA and H-CA of the problem is investigated by studying the probabilistic distribution of connections as a function of the distance, and the asymptotic size of the maximum cliques.