SBIBD(4k2, 2k2 +k,k2 +k) and Hadamard matrices of order 4k2 with maximal excess are equivalent

We show that anSBIBD(4k2, 2k2 +k,k2 +k) is equivalent to a regular Hadamard matrix of order 4k2 which is equivalent to an Hadamard matrix of order 4k2 with maximal excess.We find many newSBIBD(4k2, 2k2 +k,k2 +k) including those for evenk when there is an Hadamard matrix of order 2k (in particular all 2k ≤ 210) andk ∈ {1, 3, 5,..., 29, 33,..., 41, 45, 51, 53, 61,..., 69, 75, 81, 83, 89, 95, 99, 625, 32m, 25⋅32m,m ≥ 0}.

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