Complete OOA / Dual of NRT-Code with Only One Code Word
A linear ordered orthogonal array OOA(bsT, s,Fb, T , sT ) exists for all T ≥ 1 and s ≥ 1. Its dual is a linear [(s, T ), 0, sT + 1]-NRT-code over Fb.
The OOA (Complete OOA)
The complete OOA A = Fb(s,T ) consists of all possible runs in Fb(s,T ). Therefore it has bsT runs and the greatest possible strength sT . Any regular (s, T )×sT -matrix over Fb is a generator matrix of A, e.g. the identity matrix IsT. 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} ⊂ Fb(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,Fb, 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: 2024-09-05.
http://mint.sbg.ac.at/desc_OComplete.html