## Repetition NRT-Code with Arbitrary Length

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

### Construction of the Linear NRT-Code

The code C is constructed as

*x*,…,

*x*| ⋯ |

*x*,…,

*x*) :

*x*∈

**F**

_{b}} ⊂

**F**

_{b}

^{(s,T )}.

Therefore the 1×(*s*, *T *) matrix (1,…, 1 | ⋯ | 1,…, 1) is a generator matrix of C. For *s* > 1 C can also be constructed as an *s*-times juxtaposition or NRT-code repetition of the [(1, *T *), 1, *T *]-NRT-repetition code.

### Construction of the Orthogonal Array

The dual orthogonal array A = C^{⊥} is a hyperplane in **F**_{b}^{(s,T )} containing no coordinate axis.

### Optimality

NRT-repetition-codes meet the Singleton bound with equality and are therefore MDS-NRT-codes. Alternatively, their dual OOAs are OOAs with index unity.

### 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. “Repetition NRT-Code with Arbitrary Length.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2008-04-04.
http://mint.sbg.ac.at/desc_ORepetition-inf.html