## 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*âˆ’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 AÊ¹ 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*âˆ’1 or by dropping one row from the generator matrix.

### See Also

### 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: 2024-09-05.
http://mint.sbg.ac.at/desc_CDuplication.html