## Discarding Factors / Shortening the Dual Code

Every (linear) orthogonal array OA(*M*, *s*, *S*_{b}, *k*) yields a (linear) OA(*M*, *s* â€“ 1, *S*_{b}, *k*). Correspondingly, every linear [*s*, *n*, *d*]-code with *n* > 0 yields a linear [*s*âˆ’1, *n*âˆ’1, *d*]-code over the same field.

### The Construction for Orthogonal Arrays

The new orthogonal array AÊ¹ is obtained by discarding an arbitrary factor from the original array A, i.e, by

*x*

_{i})

_{i=2,â€¦, s}Â : Â

*âˆˆ A}.*

**x**If A is linear, a generator matrix of AÊ¹ is obtained by dropping an arbitrary column from the generator matrix of A.

### The Construction for Linear Codes

In the context of linear codes, this propagation rule corresponds to shortening a code: If C is the original code, the new code CÊ¹ is given by

*x*

_{i})

_{i=2,â€¦, s}Â : Â

*âˆˆ CÂ withÂ*

**x***x*

_{1}= 0},

i.e., it is obtained by selecting only code words with a zero at a fixed coordinate and removing this coordinate. Since all remaining code words have their weights unchanged, the minimum weight (and therefore the minimum distance) of CÊ¹ is unaffected. In the fortuitous case that *x*_{1} = 0 for all * x* âˆˆ C, CÊ¹ is actually an [

*s*âˆ’1,

*n*,

*d*]-code and the final [

*s*âˆ’1,

*n*âˆ’1,

*d*]-code can be obtained by taking a subcode.

### See Also

### References

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

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

[3] | JÃ¼rgen 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. “Discarding Factors / Shortening the Dual Code.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2024-09-05.
http://mint.sbg.ac.at/desc_CSRed.html