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Construction of Symmetric Balanced Squares with Blocksize More than One

机译：块大小不小于1的对称平衡方的构造

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摘要

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 x 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 [ks~2/v] or [ks~2/v] 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.
机译：在本文中，我们研究了对称拉丁方的推广。 v，s边和块大小为k的对称平衡平方是{1，2 ....，v}的k个元素子集的sxs对称数组，使得每个元素都以[ks / v]或[ks / v]每行和每列的单元格。每个元素都出现在数组的[ks〜2 / v]或[ks〜2 / v]单元中。根据值s，k和v，问题自然分为三个子问题：（1）v≥ks（2）s

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