2017/08/20 by Nicholas J. Cavenagh, Cavenagh, Nicholas
Engineering · Decision Sciences · #graph theory and CDMA systems #Optimal Experimental Design Methods #Antenna Design and Optimization
paper · pdf · doi:10.48550/arxiv.1708.06058
The full n-Latin square is the n\× n array with symbols 1,2,\…\n,n in each cell. In this paper we show, as part of a more general result, that\nany defining set for the full n-Latin square has size n3(1-o(1)). The full\ndesign N(v,k) is the unique simple design with parameters (v,k,v-2 choose\nk-2); that is, the design consisting of all subsets of size k from a set of\nsize v. We show that any defining set for the full design N(v,k) has size\nv choose k(1-o(1)) (as v-k becomes large). These results improve existing\nresults and are asymptotically optimal. In particular, the latter result solves\nan open problem given in (Donovan, Lefevre, et al, 2009), in which it is\nconjectured that the proportion of blocks in the complement of a full design\nwill asymptotically approach zero.\n