## Discarding Factors / Shortening the Dual Code

Every (linear) orthogonal array OA(M, s, Sb, k) yields a (linear) OA(M, s â€“ 1, Sb, k). Correspondingly, every linear [s, n, d]-code with n > 0 yields a linear [sâˆ’1, nâˆ’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

AÊ¹ := {(xi)i=2,â€¦, sÂ  : Â x âˆˆ A}.

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

{(xi)i=2,â€¦, sÂ  : Â x âˆˆ CÂ withÂ x1 = 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 x1 = 0 for all x âˆˆ C, CÊ¹ is actually an [sâˆ’1, n, d]-code and the final [sâˆ’1, nâˆ’1, d]-code can be obtained by taking a subcode.

• Generalization for arbitrary OOAs

• Corresponding result for nets and sequences

• [1, SectionÂ 1.9(VI)], [2, SectionÂ 4.4], or [3, TheoremÂ 5.1]

### 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)