Repetition NRT-Code with Arbitrary Length
A linear [(s, T ), 1, sT ]-NRT-code C over Fb exists for all T ≥ 1 and s ≥ 1. Its dual A = C⊥ is a linear ordered orthogonal array OOA(bsT−1, s,Fb, 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 Fb(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: 2024-09-05.
http://mint.sbg.ac.at/desc_ORepetition-inf.html