Construction of Symmetric Balanced Squares with Blocksize More than One

In this paper we study a generalization of symmetric latin squares. A symmetric balanced square of order v, side s and blocksize k is an s×s symmetric array of k-element subsets of {1,2,..., v} such that every element occurs in ⌊ ks/v ⌋ or ⌈ ks/v ⌉ cells of each row and column. every element occurs in ⌊ ks2/v ⌋ or ⌈ ks2v ⌉ cells of the array. Depending on the values s, k and v, the problem naturally divides into three subproblems: (1) v≥ks (2) s < v < ks (3) v ≤ s. We completely solve the first problem and we recursively reduce the third problem to the first two. For s ≤ 4 we provide direct constructions for the second problem. Moreover, we provide a general construction method for the second problem utilizing flows in a network. We have been able to show the correctness of this construction for k ≤ 3. For k≥4, the problem remains open.