Constructions for Multichannel Conflict-Avoiding Codes with AM-OPPTS Restriction

A multichannel conflict-avoiding code (MC-CAC) is a collection of two-dimensional codewords represented by zero-one matrices, and any pair of distinct codewords have at most one overlapping 1 regardless of the relative time offsets. Such codes are of practical interest as they are able to provide a hard guarantee that each active user has a successful transmission within every consecutive $n$ time slots in a wireless network with multiple asynchronous collision channels. In a more practical setting, it is assumed that in each time slot each source node can only pick one channel and send one packet in the chosen channel, and hence the at most one-packet per time slot (AM-OPPTS) restriction is appended to an MC-CAC. Only upper bounds for the number of codewords of an AM-OPPTS MC-CAC with weights three and four were known in the literature, and constructions for AM-OPPTS MC-CACs have not been explored and analyzed systematically. This paper is devoted to establishing combinatorial constructions for AM-OPPTS MC-CACs by introducing holey group divisible packings with prescribed automorphism groups. As applications of our constructions, the exact values of the sizes of both an optimal AM-OPPTS MC-CAC $(m,n,3)$ and an optimal AM-OPPTS MC-CAC $\left({m,p^{r},\frac {p+1}{2}}\right)$ are determined for certain $m,n,p$ and $r$ .