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The necessary conditions for the existence of a balanced incomplete block design on v points, with index λ and block size k, are that λ(v−1)≡0 mod (k−1), λv(v−1)≡0 mod k(k−1) . In this paper we study k=9 with λ=3, 6 and 12. We show that these conditions on v are sufficient, with the possible exceptions of v=177, 345, 385 when λ=3, and v=213 when λ=6. We give constructions of TD3(10,n)s with 13 possible exceptions, namely n∈{5,6,14,20,35,45,55,56,60,78,84,85,102}; we also reduce the number of unknown TDs with block sizes 8 and 9. |
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