Design Theory, 2nd Edition

6

Mutually Orthogonal Latin Squares

 

6.1 Introduction

Two latin squares L ₁ and L ₂ of order n are said to be orthogonal if for each ( x , y ) ∈ { 1 , 2 , . . . , n } × { 1 , 2 , . . . , n } there is exactly one ordered pair ( i , j ) such that cell ( i , j ) of L ₁ contains the symbol x and cell ( i , j ) of L ₂ contains the symbol y. In other words, if L ₁ and L ₂ are superimposed, the resulting set of n ² ordered pairs are distinct. The latin squares L ₁, L ₂, . . ., L _t are said to be mutually orthogonal if for 1 ≤ a ≠ b ≤ t, L _a and L _b are orthogonal.

The easiest way to show that a pair of latin squares are orthogonal is to use the “ famous” two -finger rule.

The Two-Finger Rule Let L and M be latin squares ...