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Cyclic (v; r, s; λ) Difference Families with Two Base Blocks and v ≤ 50
Dragomir Z. Dokovic
Department of Pure Mathematics, University ofWaterloo,Waterloo, Ontario, N2L 3G1, Canada
djokovic@uwaterloo.ca
Annals of Combinatorics 15 (2) pp.233-254 April, 2011
AMS Subject Classification: 05B20, 05B30
Abstract:
We construct many new cyclic (v; r, s; λ) difference families with v ≥ 2r ≥ 2s ≥ 4 and v ≤ 50. In particular, we construct the difference families with parameters (45; 18, 10; 9), (45; 22, 22; 21), (47; 21, 12; 12), (47; 19, 15; 12), (47; 22, 14; 14), (48; 20, 10; 10), (48; 24, 4; 12), (50; 25, 20; 20), for which the existence question was an open problem. We point out that the (45; 22, 22; 21) difference family gives a balanced incomplete block design (BIBD) with parameters v = 45, b = 90, r = 44, k = 22, and λ = 21, and that the one with parameters (50; 25, 20; 20) gives a pair of binary sequences of length 50 with zero periodic autocorrelation
function (the periodic analog of a Golay pair). The new SDSs include nine new D-optimal designs. A normal form for cyclic difference families (with base blocks of arbitrary sizes) is proposed and used effectively in compiling our selective listings in Tables 3–6 of known and new difference families in the above range.
Keywords: difference family, supplementary difference sets, balanced incomplete block designs, genetic algorithm

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