## (*u*, *u*+*v*)-Construction

Given two (linear) orthogonal arrays OA(*M*_{i}, *s*_{i}, *S*_{b}, *k*_{i}) with *s*_{1} ≤ *s*_{2}, a (linear) OA(*M*_{1}*M*_{2}, *s*_{1} + *s*_{2}, *S*_{b}, *k*) with *k* = min{2*k*_{1} +1, *k*_{2}} can be constructed.

Correspondingly, given two (linear) (*s*_{i}, *N*_{i}, *d*_{i})-codes over the same field with *s*_{1} ≤ *s*_{2}, a (linear) (*s*_{1} + *s*_{2}, *N*_{1}*N*_{2}, *d*) with *d* = min{2*d*_{1}, *d*_{2}} can be constructed.

### The Construction for Orthogonal Arrays

Let A_{1} and A_{2} denote the orthogonal arrays with parameters OA(*M*_{1}, *s*_{1}, *S*_{b}, *k*_{1}) and OA(*M*_{2}, *s*_{2}, *S*_{b}, *k*_{2}), respectively, and let *s*_{1} ≤ *s*_{2}. Then the resulting orthogonal array A with parameters OA(*M*_{1}*M*_{2}, *s*_{1} + *s*_{2}, *S*_{b}, *k*) is given by

*–*

**u***π*(

*),*

**v***) :*

**v***∈ A*

**u**_{1},

*∈ A*

**v**_{2}}

where *π* : *S*_{b}^{s2}→*S*_{b}^{s1} is the projection selecting the first *s*_{1} coordinates.

If A_{1} and A_{2} are linear with *M*_{i} = *b*^{mi} and *m*_{i}×*s*_{i} generator matrices **H**_{i}, the generator matrix of A is given by

with *π*(**H**_{2}) denoting the first *s*_{1} columns of **H**_{2}.

### The Construction for Codes

Let C_{1} be an (*s*_{1}, *N*_{1}, *d*_{1})-code and let C_{2} be an (*s*_{2}, *N*_{2}, *d*_{2})-code, both over **F**_{b} with *s*_{1} ≤ *s*_{2}. Then it is shown in [1] that the set of vectors

*,(*

**u***, 0*

**u**_{n1×(s2-s1)}) +

*) :*

**v***∈ C*

**u**_{1},

*∈ C*

**v**_{2}}

is an (*s*_{1} + *s*_{2}, *N*_{1}*N*_{2}, *d*)-code over **F**_{b} with *d* = min{2*d*_{1}, *d*_{2}}. The result for *s*_{1} = *s*_{2} can already be found in [2].

If C_{1} and C_{2} are linear with *N*_{i} = *b*^{ni}, *m*_{i} = *s*_{i} – *n*_{i}, *n*_{i}×*s* generator matrices **G**_{i}, and *m*_{i}×*s*_{i} parity check matrices **H**_{i}, the generator matrix of the new linear [*s*_{1} + *s*_{2}, *n*_{1} + *n*_{2}, *d*]-code C is given by

its parity check matrix is shown above.

### See Also

Generalization for OOAs

Corresponding result for nets

If

*b*> 2, this result can be generalized such that more than two codes are used, leading to the (*u*,*u*−*v*,*u*+*v*+*w*)-construction and to the generalized (*u*,*u*+*v*)-construction.A weaker construction yielding codes and OAs of the same size, but with smaller minimum distance / strength is the direct product, which can be seen as a “(

*u*,*v*)-construction”.[3, Section 2.9], [4, Section 4.4], [5, Section 10.3], or [6, Theorem 5.10]

### References

[1] | Neil J. A. Sloane and D. S. Whitehead. A new family of single-error correcting codes. IEEE Transactions on Information Theory, 16(6):717–719, November 1970. |

[2] | Morris Plotkin. Binary codes with specified minimum distance. IEEE Transactions on Information Theory, 6(4):445–450, September 1960. |

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

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

[5] | A. S. Hedayat, Neil J. A. Sloane, and John Stufken.Orthogonal Arrays.Springer Series in Statistics. Springer-Verlag, 1999. |

[6] | 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. “(*u*, *u*+*v*)-Construction.”
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Version: 2015-09-03.
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