OOA with Only One Run / Dual of NRT-Code Without Redundancy

A linear ordered orthogonal array OOA(1, s, S_b, T , 0) which is the dual of a linear [(s, T ), sT , 1]-NRT-code exists for arbitrary values of s.

The OOA (Trivial OOA)

The orthogonal array A = { 0} ⊂ F_b^{(s,T )} is a linear OOA(1, s,F_b, T , 0) for arbitrary values of s. It has only a single run 0 and therefore strength k = 0. Thus it is called trivial OOA or OOA with only one run. The generator matrix of A is an 0×(s, T ) matrix. If interpreted as a linear NRT-code, A is an [(s, T ), 0, Ts + 1]-trivial NRT-code.

Its Dual Code (NRT-Code Without Redundancy)

The dual code C = A^⊥ is a linear [(s, T ), sT , 1]-code over F_b, namely the NRT-code without redundancy or complete NRT-code. It consists of all code words in F_q^{(s,T )}, i.e., C = F_b^{(s,T )}. Since every possible vector from F_b^{(s,T )} is a valid code word, C has no redundancy and cannot even detect a single error. Any regular sT ×(s, T )-matrix over F_b is a generator matrix of C, in particular the sT ×(s, T ) identity matrix. If interpreted as a linear OOA, C is an OOA(b^sT, s,F_b, T , sT ), the complete OOA.

Optimality

A trivial OOA meets the trivial lower bound on t and the Rao bound with equality and is therefore an OOA with index unity as well as a tight OOA. Alternatively a code without redundancy is an MDS-NRT-code as well as a perfect NRT-code.

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. “OOA with Only One Run / Dual of NRT-Code Without Redundancy.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2024-09-05. http://mint.sbg.ac.at/desc_OTrivial.html