Concatenation of Two Codes
Given a linear [s1, n1, d1]-code C1 over Fbn2 and a linear [s2, n2, d2]-code C2 over Fb, a linear [s1s2, n1n2, d1d2]-code over Fb can be constructed. Using duality, a linear orthogonal array OA(bs1s2-n1n2, s1s2,Fb, d1d2 − 1) can be constructed based on a linear OA(bs1-m1, s1,Fbs2-n2, d1 − 1) and OA(bs2-m2, s2,Fb, d2 − 1).
Construction
Let φ : Fbn2↔C2 ⊆ Fbs2 denote an arbitrary Fb-linear bijection. The new code C is given by
Special Cases
A number of other propagation rules can be interpreted as concatenation with special codes:
Repeating each code word of C1 over Fb1 s2 times can be accomplished by using an [s2, 1, s2]-repetition code over Fb as C2.
Repeating each code word of C2 over Fb s1 times can be accomplished by using an [s1, 1, s1]-repetition code over Fbn2 as C1.
The trace code from Fbn2 to Fb of C1 is obtained by using the [n2, n2, 1]-code without redundancy as C2.
See Also
Generalization for arbitrary NRT-codes
[1, Theorem 5.9]
References
| [1] | Jürgen Bierbrauer. Introduction to Coding Theory. Discrete Mathematics and its Applications. Chapman & Hall/CRC, Boca Raton, London, New York, Washington D.C., 2004. MR2079734 (2005f:94001) |
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. “Concatenation of Two Codes.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2015-09-03.
http://mint.sbg.ac.at/desc_CConcatD.html