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

A linear ordered orthogonal array OOA(*b*^{sT}, *s*,**F**_{b}, *T *, *sT *) exists for all *T * ≥ 1 and *s* ≥ 1. Its dual is a linear [(*s*, *T *), 0, *sT * + 1]-NRT-code over **F**_{b}.

### The OOA (Complete OOA)

The complete OOA A = **F**_{b}^{(s,T )} consists of all possible runs in **F**_{b}^{(s,T )}. Therefore it has *b*^{sT} runs and the greatest possible strength *sT *. Any regular (*s*, *T *)×*sT *-matrix over **F**_{b} is a generator matrix of A, e.g. the identity matrix **I**_{sT}. 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} ⊂ **F**_{b}^{(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*,**F**_{b}, *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.

### Copyright

Copyright © 2004, 2005, 2006, 2007, 2008, 2009, 2010 by Rudolf Schürer and Wolfgang Ch. Schmid.

Cite this as: Rudolf Schürer and Wolfgang Ch. Schmid. “Complete OOA / Dual of NRT-Code with Only One Code Word.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2008-04-04.
http://mint.sbg.ac.at/desc_OComplete.html