Code Embedding in Larger Space
Every (linear) orthogonal array OA(bm, s, Sb, k) yields a (linear) OA(bm+1, s + 1, Sb, k). Correspondingly, every (linear) (s, N, d)-code yields a (linear) (s + 1, N, d)-code over the same field.
For linear codes over ℤ2 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
If H is a generator matrix of A, the generator matrix of Aʹ is given by



In other words, Aʹ is the direct product of A and the complete orthogonal array OA(b, 1, Sb, 1).
Construction for Linear Codes
The new code Cʹ is obtained by embedding C ⊆ Fbs in Fbs+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: 2024-09-05.
http://mint.sbg.ac.at/desc_CEmbeddingInLargerSpace.html