## Concatenation of Two Codes

Given a linear [*s*_{1}, *n*_{1}, *d*_{1}]-code C_{1} over **F**_{bn2} and a linear [*s*_{2}, *n*_{2}, *d*_{2}]-code C_{2} over **F**_{b}, a linear [*s*_{1}*s*_{2}, *n*_{1}*n*_{2}, *d*_{1}*d*_{2}]-code over **F**_{b} can be constructed. Using duality, a linear orthogonal array OA(*b*^{s1s2-n1n2}, *s*_{1}*s*_{2},**F**_{b}, *d*_{1}*d*_{2} − 1) can be constructed based on a linear OA(*b*^{s1-m1}, *s*_{1},**F**_{bs2-n2}, *d*_{1} − 1) and OA(*b*^{s2-m2}, *s*_{2},**F**_{b}, *d*_{2} − 1).

### Construction

Let *φ* : **F**_{bn2}↔C_{2} ⊆ **F**_{b}^{s2} denote an arbitrary **F**_{b}-linear bijection. The new code C is given by

*φ*(

*x*

_{i}))

_{i=1,…, s1}:

*∈ C*

**x**_{1}).

### Special Cases

A number of other propagation rules can be interpreted as concatenation with special codes:

Repeating each code word of C

_{1}over**F**_{b1}*s*_{2}times can be accomplished by using an [*s*_{2}, 1,*s*_{2}]-repetition code over**F**_{b}as C_{2}.Repeating each code word of C

_{2}over**F**_{b}*s*_{1}times can be accomplished by using an [*s*_{1}, 1,*s*_{1}]-repetition code over**F**_{bn2}as C_{1}.The trace code from

**F**_{bn2}to**F**_{b}of C_{1}is obtained by using the [*n*_{2},*n*_{2}, 1]-code without redundancy as C_{2}.

### 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