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

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: 2015-09-03. http://mint.sbg.ac.at/desc_ORepetition-inf.html