Complete OA / Dual of Code with Only One Code Word

A linear orthogonal array OA(bs, s,Fb, s) exists for all s ≥ 1. Its dual is a linear [s, 0, s + 1]-code over Fb.

The Orthogonal Array (Complete OA)

The complete OA A = Fbs consists of all possible runs in Fbs. Therefore it has bs runs and the greatest possible strength s. Any regular s×s-matrix over Fb is a generator matrix of A, e.g. the s×s identity matrix Is. When interpreted as a linear code, A is the [s, s, 1]-code without redundancy.

Its Dual Code (Trivial Code with Only One Code Word)

The dual code of A is C = A = { 0} ⊂ Fbs, i.e., it contains only the zero vector, and is therefore called trivial code. The s matrix is as generator matrix of C. C can also be obtained by shortening an [s + 1, 1, s + 1]-repetition code. When interpreted as an orthogonal array, C is an OA(1, s,Fb, 0), namely the trivial orthogonal array with only one run.

Optimality

A trivial code meets the Singleton bound with equality and is therefore an MDS-code. Alternatively, a complete OA is an OA with index unity.