## Complete OOA / Dual of NRT-Code with Only One Code Word

A linear ordered orthogonal array OOA(bsT, s,Fb, T , sT ) exists for all T ≥ 1 and s ≥ 1. Its dual is a linear [(s, T ), 0, sT + 1]-NRT-code over Fb.

### The OOA (Complete OOA)

The complete OOA A = Fb(s,T ) consists of all possible runs in Fb(s,T ). Therefore it has bsT runs and the greatest possible strength sT . Any regular (s, T sT -matrix over Fb is a generator matrix of A, e.g. the identity matrix IsT. When interpreted as a linear NRT-code, A is the [(s, T ), sT , 1]-code without redundancy.

### Its Dual NRT-Code (Trivial NRT-Code with Only One Code Word)

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

### Optimality

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