Complete OA / Dual of Code with Only One Code Word

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

The Orthogonal Array (Complete OA)

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

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

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

Optimality

A trivial code meets the Singleton bound with equality and is therefore an MDS-code. Alternatively, a complete OA is an OA 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 OA / Dual of 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_CComplete.html