Latin squares

A latin square of order is an array in which each of symbols occur once in each row and once in each column. Typically, the symbol in row column is denoted .

Let . The array defined by where is a latin square.

Proof:
Given any row and symbol we can let and then get that , and we know that for some , so the symbol will be in cell . Same goes for any column, so it is indeed a latin square.

Two latin squares of the same order are orthogonal if for every ordered pair of symbols there is exactly one position such that and .

Let be prime and be nonzero. Then where defines a latin square. Furthermore, if is nonzero and , then and are orthogonal.

Proof:
Since is InvertibleInvertible, it follows from the proof of TheoremTheorem that is a latin square. Furthermore, given a pair we have the system of equations where we used that meaning and , meaning both of these are invertible.