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- W2963266952 abstract "The full n-Latin square is the $$ntimes n$$ array with symbols $$1,2,ldots ,n$$ in each cell. In this paper we show, as part of a more general result, that any defining set for the full n-Latin square has size $$n^3(1-o(1))$$ . The full design N(v, k) is the unique simple design with parameters $$(v,k,{v-2 atopwithdelims ()k-2})$$ ; that is, the design consisting of all subsets of size k from a set of size v. We show that any defining set for the full design N(v, k) has size $${vatopwithdelims ()k}(1-o(1))$$ (as $$v-k$$ becomes large). These results improve existing results and are asymptotically optimal. In particular, the latter result solves an open problem given in Donovan et al. (Graphs Comb 25:825–839, 2009), in which it is conjectured that the proportion of blocks in the complement of a full design will asymptotically approach zero." @default.
- W2963266952 created "2019-07-30" @default.
- W2963266952 creator A5080083927 @default.
- W2963266952 date "2018-05-09" @default.
- W2963266952 modified "2023-09-28" @default.
- W2963266952 title "Lower Bounds on the Sizes of Defining Sets in Full n-Latin Squares and Full Designs" @default.
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- W2963266952 doi "https://doi.org/10.1007/s00373-018-1895-7" @default.
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