Bounds on Maximal Tournament Codes

In this paper, we improve the best-known upper bound on the size of maximal tournament codes, and solve the related problem of edge-covering a complete graph with a minimum number of bipartite graphs of bounded size. Tournament codes are sets of {0,1,∗} strings closely related to self-synchronizing codes. We improve the current asymptotic upper bound on the size of a length-k tournament code (given by van Lint and van Pul) by a factor in the exponent, and then demonstrate a conditional method for improving the upper bound based on the number of 0 and 1 characters in the optimal tournament code. We also consider a previously-unused equivalence between tournament codes and certain graphs, which relates tournament codes to bipartite coverings of complete graphs. We solve the problem of covering a complete graph with a minimum number of bicliques of bounded size, determining that the minimum number of bicliques of component size x needed to cover a complete graph on n vertices is Θ ((n x )2 + (n x ) logx ) (an original result). Finally, we demonstrate the limitations of applying the minimum bounded biclique covering result to the maximal tournament code problem.