A note on a combinatorial problem

The combinatorial problem under consilderatlon makes Its appearance in the study of projective planes, Hadamard matrices, and block designs. In the combinatorial problem of Todd arising in the study of Hadamard matrices it is usually assumed that X =k(k-1)/(v-1) [3; 4].1 For the symmetrical block designs each element of the arrangement is required to occur exactly k times, and it is then easy to verify that X = k(k -1 )/(v -1) [1 ]. Further theorems concerning the possibility of this combinatorial problem for a given vI k, and X may be found in [3 ]. To prove the theorem let the elements xi, , x, be listed in a row, and let the sets TF, * , T, be listed in a column. Form the incidence matrix A of the arrangement in the usual way by inserting a one in column i and row j if xi belongs to set TF, and a zero in the contrary case [2; 3]. Now let si denote the sum of column i of the