## Juxtaposition

Give an (*s*_{1}, *N*, *d*_{1})-code C_{1} and an (*s*_{2}, *N*, *d*_{2})-code C_{2}, an (*s*_{1} + *s*_{2}, *N*, *d*_{1} + *d*_{2})-code over the same field can be constructed. Therefore a linear orthogonal array OA(*b*^{s1+s2−n}, *s*_{1} + *s*_{2}, *S*_{b}, *k*_{1} + *k*_{2} + 1) can be constructed from a linear OA(*b*^{s1−n}, *s*_{1}, *S*_{b}, *k*_{1}) and a linear OA(*b*^{s2−n}, *s*_{2}, *S*_{b}, *k*_{2}).

### Construction for Linear Codes

Let C_{1} = {**x**_{1},…,**x**_{N}} and C_{2} = {**y**_{1},…**y**_{N}}, then the new code C is given by

**x**_{i},

**y**_{i}) :

*i*= 1,…,

*N*}.

If the C_{i} are linear [*s*_{i}, *n*, *d*_{i}]-codes with generator matrices **G**_{i}, then C is an [*s*_{1} + *s*_{2}, *n*, *d*_{1} + *d*_{2}]-code with generator matrix

If **G**_{i} = (**I**_{n}**A**_{i}) and **H**_{i} = (**B**_{i}**I**_{mi}) with **B**_{i} = – **A**_{i}^{T } and *m*_{i} = *s*_{i}−*n* is a parity check matrix of *C*_{i}, then the generator matrix of *C* is given by

and its parity check matrix by

see ([1, Problem 10.1]).

### See Also

Generalization for arbitrary OOAs

Juxtaposition is a special case of construction X4

[2, Section 1.3, Problem (17)], [2, Figure 2.6], or [1, Section 10.2].

### References

[1] | A. S. Hedayat, Neil J. A. Sloane, and John Stufken.Orthogonal Arrays.Springer Series in Statistics. Springer-Verlag, 1999. |

[2] | F. Jessie MacWilliams and Neil J. A. Sloane.The Theory of Error-Correcting Codes.North-Holland, Amsterdam, 1977. |

### 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. “Juxtaposition.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2015-09-03.
http://mint.sbg.ac.at/desc_CJuxtaposition.html