Truncation

Every (linear) orthogonal array OA(bm, s, Sb, k) with k > 0 yields a (linear) OA(bm−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

Aʹ = {(xi)i=2,…, s  :  x ∈ A and x1 = 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 = $\displaystyle \left(\vphantom{\begin{array}{cc} \vec{0}_{(m−1)\times1} & \vec{H}ʹ\\ x & \vec{y}\end{array}}\right.$$\displaystyle \begin{array}{cc} \vec{0}_{(m−1)\times1} & \vec{H}ʹ\\ x & \vec{y}\end{array}$$\displaystyle \left.\vphantom{\begin{array}{cc} \vec{0}_{(m−1)\times1} & \vec{H}ʹ\\ x & \vec{y}\end{array}}\right)$

(which can always be achieved using elementary row operations), the (m – 1)×(s−1)-matrix Hʹ 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

Cʹ = {(xi)i=2,…, s  :  x ∈ C}.

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: 2024-09-05. http://mint.sbg.ac.at/desc_CMRedS.html

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