## OA with Only One Run / Dual of Code Without Redundancy

A linear orthogonal array OA(1, *s*, *S*_{b}, 0) which is the dual of a linear [*s*, *s*, 1]-code exists for arbitrary values of *s*.

### The Orthogonal Array (Trivial OA)

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

### Its Dual Code (Code Without Redundancy)

The dual code C = A^{⊥} is a linear [*s*, *s*, 1]-code over **F**_{b}, namely the code without redundancy or complete code. It consists of all code words in **F**_{q}^{s}, i.e., C = **F**_{b}^{s}. Since every possible vector from **F**_{b}^{s} is a valid code word, C has no redundancy and cannot even detect a single error. Any regular *s*×*s*-matrix over **F**_{b} is a generator matrix of C, in particular the *s*×*s* identity matrix. If interpreted as a linear orthogonal array, C is an OA(*b*^{s}, *s*,**F**_{b}*s*), the complete OA.

### Optimality

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

### See Also

Generalization for arbitrary OOAs

### 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. “OA with Only One Run / Dual of Code Without Redundancy.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2008-04-04.
http://mint.sbg.ac.at/desc_CTrivial.html