Duplication / Taking a Subcode of the Dual Code

Every (linear) orthogonal array OA(b^m, s, S_b, k) yields a (linear) OA(b^m+1, s, S_b, k). Correspondingly, every linear [s, n, d]-code with n > 0 yields a linear [s, n−1, d]-code.

The same result can always be obtained using a combination of discarding factors / shortening and embedding in a larger space.

Construction for Orthogonal Arrays

The new orthogonal array Aʹ is obtained by replicating each run of the original array A b times or by appending an arbitrary row to the generator matrix of A. If A is simple, linear, and m < s, a new orthogonal array without duplicate runs can be obtained by appending one row to the generator matrix such that the resulting matrix has full rank.

Construction for Linear Codes

The new code is obtained simply by taking an arbitrary subspace of dimension n−1 or by dropping one row from the generator matrix.

See Also

• Generalization for arbitrary OOAs

• Corresponding result for nets

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. “Duplication / Taking a Subcode of the Dual Code.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03. http://mint.sbg.ac.at/desc_CDuplication.html