## OOA with Only One Run / Dual of NRT-Code Without Redundancy

A linear ordered orthogonal array OOA(1, s, Sb, T , 0) which is the dual of a linear [(s, T ), sT , 1]-NRT-code exists for arbitrary values of s.

### The OOA (Trivial OOA)

The orthogonal array A = { 0} ⊂ Fb(s,T ) is a linear OOA(1, s,Fb, T , 0) for arbitrary values of s. It has only a single run 0 and therefore strength k = 0. Thus it is called trivial OOA or OOA with only one run. The generator matrix of A is an 0×(s, T ) matrix. If interpreted as a linear NRT-code, A is an [(s, T ), 0, Ts + 1]-trivial NRT-code.

### Its Dual Code (NRT-Code Without Redundancy)

The dual code C = A is a linear [(s, T ), sT , 1]-code over Fb, namely the NRT-code without redundancy or complete NRT-code. It consists of all code words in Fq(s,T ), i.e., C = Fb(s,T ). Since every possible vector from Fb(s,T ) is a valid code word, C has no redundancy and cannot even detect a single error. Any regular sT ×(s, T )-matrix over Fb is a generator matrix of C, in particular the sT ×(s, T ) identity matrix. If interpreted as a linear OOA, C is an OOA(bsT, s,Fb, T , sT ), the complete OOA.

### Optimality

A trivial OOA meets the trivial lower bound on t and the Rao bound with equality and is therefore an OOA with index unity as well as a tight OOA. Alternatively a code without redundancy is an MDS-NRT-code as well as a perfect NRT-code.