## Code Embedding in Larger Space

Every (linear) orthogonal array OA(*b*^{m}, *s*, *S*_{b}, *k*) yields a (linear) OA(*b*^{m+1}, *s* + 1, *S*_{b}, *k*). Correspondingly, every (linear) (*s*, *N*, *d*)-code yields a (linear) (*s* + 1, *N*, *d*)-code over the same field.

For linear codes over ℤ_{2} and odd minimum distance an even stronger propagation rule (Adding a Parity Check Bit) is available.

### Construction for Orthogonal Arrays

Based on a given orthogonal A the new orthogonal array Aʹ is obtained as

*,*

**x***y*) :

*∈ A,*

**x***y*∈

*S*}.

If * H* is a generator matrix of A, the generator matrix of Aʹ is given by

*ʹ = .*

**H**In other words, Aʹ is the direct product of A and the complete orthogonal array OA(*b*, 1, *S*_{b}, 1).

### Construction for Linear Codes

The new code Cʹ is obtained by embedding C ⊆ **F**_{b}^{s} in **F**_{b}^{s+1}. In other words, Cʹ is constructed by appending a 0 to every code word of C or by appending an all-zero column to the generator matrix of C.

### 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. “Code Embedding in Larger Space.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2008-04-04.
http://mint.sbg.ac.at/desc_CEmbeddingInLargerSpace.html