## 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  :  xA 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 =   (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  :  xC}.

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.

• Generalization for arbitrary OOAs

• [1, Section 1.9(II)], [2, Section 4.4], or [3, Theorem 5.2]

### 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)