## Discarding Factors / Shortening the Dual Code

Every (linear) orthogonal array OA(M, s, Sb, k) yields a (linear) OA(M, s – 1, Sb, k). Correspondingly, every linear [s, n, d]-code with n > 0 yields a linear [s−1, n−1, d]-code over the same field.

### The Construction for Orthogonal Arrays

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

Aʹ := {(xi)i=2,…, s  :  xA}.

If A is linear, a generator matrix of Aʹ 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 Cʹ is given by

{(xi)i=2,…, s  :  xC with 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 Cʹ is unaffected. In the fortuitous case that x1 = 0 for all xC, Cʹ is actually an [s−1, n, d]-code and the final [s−1, n−1, d]-code can be obtained by taking a subcode.

• Generalization for arbitrary OOAs

• Corresponding result for nets and sequences

• [1, Section 1.9(VI)], [2, Section 4.4], or [3, Theorem 5.1]

### References

  F. Jessie MacWilliams and Neil J. A. Sloane.The Theory of Error-Correcting Codes.North-Holland, Amsterdam, 1977.  Jacobus H. van Lint.Introduction to Coding Theory, volume 86 of Graduate Texts in Mathematics.Springer-Verlag, second edition, 1991.  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)