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

Aʹ = {(x, y)  :  x ∈ A, y ∈ S}.

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

Hʹ = $\displaystyle \left(\vphantom{\begin{array}{cc} \vec{H} & \vec{0}_{b^{m}\times1}\\ \vec{0}_{1\times s} & 1\end{array}}\right.$$\displaystyle \begin{array}{cc} \vec{H} & \vec{0}_{b^{m}\times1}\\ \vec{0}_{1\times s} & 1\end{array}$$\displaystyle \left.\vphantom{\begin{array}{cc} \vec{H} & \vec{0}_{b^{m}\times1}\\ \vec{0}_{1\times s} & 1\end{array}}\right)$.

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

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

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