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Orthogonal trades in complete sets of MOLS

Orthogonal trades in complete sets of MOLS

##### Abstract

Let Bₚ be the Latin square given by the addition table for the integers modulo an odd prime p (i.e. the Cayley table for (Zₚ, +)). Here we consider the properties of Latin trades in Bₚ which preserve orthogonality with one of the p−1 MOLS given by the finite field construction. We show that for certain choices of the orthogonal mate, there is a lower bound logarithmic in p for the number of times each symbol occurs in such a trade, with an overall lower bound of (log p)² / log log p for the size of such a trade. Such trades imply the existence of orthomorphisms of the cyclic group which differ from a linear orthomorphism by a small amount. We also show that any transversal in Bₚ hits the main diagonal either p or at most p − log₂ p – 1 times. Finally, if p ≡ 1 (mod 6) we show the existence of a Latin square which is orthogonal to Bₚ and which contains a 2 × 2 subsquare.

##### Type

Journal Article

##### Type of thesis

##### Series

##### Citation

Cavenagh, N. J., Donovan, D. M., & Demirkale, F. (2017). Orthogonal trades in complete sets of MOLS. The Electronic Journal of Combinatorics, 24(3).

##### Date

2017

##### Publisher

The Electronic Journal of Combinatorics