Complete OOA / Dual of NRT-Code with Only One Code Word

A linear ordered orthogonal array OOA(b^sT, s,F_b, T , sT ) exists for all T ≥ 1 and s ≥ 1. Its dual is a linear [(s, T ), 0, sT + 1]-NRT-code over F_b.

The OOA (Complete OOA)

The complete OOA A = F_b^{(s,T )} consists of all possible runs in F_b^{(s,T )}. Therefore it has b^sT runs and the greatest possible strength sT . Any regular (s, T )×sT -matrix over F_b is a generator matrix of A, e.g. the identity matrix I_sT. When interpreted as a linear NRT-code, A is the [(s, T ), sT , 1]-code without redundancy.

Its Dual NRT-Code (Trivial NRT-Code with Only One Code Word)

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

Optimality

A trivial NRT-code meets the Singleton bound with equality and is therefore an MDS-NRT-code. Alternatively, the complete OOA is an OOA 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 OOA / Dual of NRT-Code with Only One Code Word.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03. http://mint.sbg.ac.at/desc_OComplete.html