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It is well known that mutually orthogonal latin squares, or MOLS, admit a(Kronecker) product construction. We show that, under mild conditions, `tripleproducts' of MOLS can result in a gain of one square. In terms of transversaldesigns, the technique is to use a construction of Rolf Rees twice: once toobtain a coarse resolution of the blocks after one product, and next toreorganize classes and resolve the blocks of the second product. Asconsequences, we report a few improvements to the MOLS table and obtain aslight strengthening of the famous theorem of MacNeish.

A three-factor product construction for mutually orthogonal latin squares