OA with Only One Run / Dual of Code Without Redundancy

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

The Orthogonal Array (Trivial OA)

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

Its Dual Code (Code Without Redundancy)

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

Optimality

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

See Also

• Generalization for arbitrary OOAs

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. “OA with Only One Run / Dual of 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_CTrivial.html