## Truncation

Every (linear) orthogonal array OA(*b*^{m}, *s*, *S*_{b}, *k*) with *k* > 0 yields a (linear) OA(*b*^{m−1}, *s*−1, *b*, *k*−1). Correspondingly, every (linear) (*s*, *N*, *d*)-code with *d* > 1 yields a (linear) (*s*−1, *N*, *d*−1)-code over the same field.

### Construction for Orthogonal Arrays

The new orthogonal array Aʹ is obtained from A as

*x*

_{i})

_{i=2,…, s}:

*∈ A and*

**x***x*

_{1}= 0},

i.e., by dropping one factor and all runs that do not have a certain fixed element in this factor. If * H* is a generator matrix of A in the form

*=*

**H**(which can always be achieved using elementary row operations), the (*m* – 1)×(*s*−1)-matrix * H*ʹ is a generator matrix of Aʹ.

### Construction for Codes

The new code Cʹ is constructed by dropping one coordinate from all code words of C, i.e., as

*x*

_{i})

_{i=2,…, s}:

*∈ C}.*

**x**A generator matrix of Cʹ can be obtained by dropping an arbitrary column from a generator matrix of C.

This construction method is also known as puncturing or projecting a code.

### See Also

### References

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

[2] | Jacobus H. van Lint.Introduction to Coding Theory, volume 86 of Graduate Texts in Mathematics.Springer-Verlag, second edition, 1991. |

[3] | 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. “Truncation.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2015-09-03.
http://mint.sbg.ac.at/desc_CMRedS.html