## Complete OA / Dual of Code with Only One Code Word

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

### The Orthogonal Array (Complete OA)

The complete OA A = **F**_{b}^{s} consists of all possible runs in **F**_{b}^{s}. Therefore it has *b*^{s} runs and the greatest possible strength *s*. Any regular *s*×*s*-matrix over **F**_{b} is a generator matrix of A, e.g. the *s*×*s* identity matrix **I**_{s}. When interpreted as a linear code, A is the [*s*, *s*, 1]-code without redundancy.

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

The dual code of A is C = A^{⊥} = { 0} ⊂ **F**_{b}^{s}, i.e., it contains only the zero vector, and is therefore called trivial code. The 0×*s* matrix is as generator matrix of C. C can also be obtained by shortening an [*s* + 1, 1, *s* + 1]-repetition code. When interpreted as an orthogonal array, C is an OA(1, *s*,**F**_{b}, 0), namely the trivial orthogonal array with only one run.

### Optimality

A trivial code meets the Singleton bound with equality and is therefore an MDS-code. Alternatively, a complete OA is an OA 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 OA / Dual of 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_CComplete.html