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

Let C_{i} be (*s*_{i}, *N*_{i}, *d*_{i})-codes over **F**_{b} for *i* = 1, 2, 3 with *s*_{1} ≤ *s*_{2} ≤ *s*_{3} and *b* ≥ 3. For *i* ≤ *j* let *φ*_{i,j} : **F**_{b}^{si}→**F**_{b}^{sj} denote the embedding of **F**_{b}^{si} in **F**_{b}^{sj} defined by

*φ*((

*x*

_{1},…,

*x*

_{si})) := (

*x*

_{1},…,

*s*

_{si}, 0,…, 0)

and let *α* ∈ **F**_{b} ∖ {0, 1}. Then the set of vectors

*,*

**u***φ*

_{1,2}(

*) +*

**u***,*

**v***φ*

_{1,3}(

*) +*

**u***αφ*

_{2,3}(

*) +*

**v***) :*

**w***∈ C*

**u**_{1},

*∈ C*

**v**_{2},

*∈ C*

**w**_{3}}

is an (*s*_{1} + *s*_{2} + *s*_{3}, *N*_{1}*N*_{2}*N*_{3}, *d*)-code over **F**_{b} with *d* = min{3*d*_{1}, 2*d*_{2}, *d*_{3}}. If the C_{i} are linear with *N*_{i} = *b*^{ni}, *n*_{i}×*s*_{i} generator matrices **G**_{i}, *m*_{i} := *s*_{i} – *n*_{i}, and *m*_{i}×*s*_{i} parity check matrices **H**_{i}, the generator matrix * G* of the new linear [

*s*

_{1}+

*s*

_{2}+

*s*

_{3},

*n*

_{1}+

*n*

_{2}+

*n*

_{3},

*d*]-code C is given by

*= ,*

**G**its parity check matrix * H* by

*=*

**H**with *φ*_{i,j}(**G**_{i}) denoting the *n*_{i}×*s*_{j}-matrix (**G**_{i} 0_{ni×(sj-si)}) and *π*_{j,i}(**H**_{j}) denoting the first *s*_{i} columns from **H**_{j}.

In the context of orthogonal arrays three linear OAs A_{1}, A_{2}, and A_{3} with parameters OA(*b*^{mi}, *s*_{i},**F**_{b}, *k*_{i}) and with generator matrices **H**_{i} and *s*_{1} ≤ *s*_{2} ≤ *s*_{3} can be used for constructing an OA(*b*^{m1+m2+m3}, *s*_{1} + *s*_{2} + *s*_{3},**F**_{b}, *k*) with *k* = min{3*k*_{1} +2, 2*k*_{2} +1, *k*_{3}} and generator matrix * H*, where

*is the parity check matrix shown above.*

**H**If *b* ≥ 4, this result can be generalized such that more than three codes are used, leading to the generalized (*u*, *u* + *v*)-construction. The special case where C_{1} is the trivial [0, 0]-code is identical to the (*u*, *u* + *v*)-construction.

### 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*, *u*+*v*+*w*)-Construction.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2015-09-03.
http://mint.sbg.ac.at/desc_CUUMinusVUPlusVPlusW.html