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

A linear ordered orthogonal array OOA(1, *s*, *S*_{b}, *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} ⊂ **F**_{b}^{(s,T )} is a linear OOA(1, *s*,**F**_{b}, *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 **F**_{b}, namely the NRT-code without redundancy or complete NRT-code. It consists of all code words in **F**_{q}^{(s,T )}, i.e., C = **F**_{b}^{(s,T )}. Since every possible vector from **F**_{b}^{(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 **F**_{b} is a generator matrix of C, in particular the *sT *×(*s*, *T *) identity matrix. If interpreted as a linear OOA, C is an OOA(*b*^{sT}, *s*,**F**_{b}, *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.

### See Also

Special case for orthogonal arrays and linear codes

### 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. “OOA with Only One Run / Dual of NRT-Code Without Redundancy.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2015-09-03.
http://mint.sbg.ac.at/desc_OTrivial.html