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

The following generalization of the (*u*, *u* + *v*)-construction is a special case of Blokh-Zyablov-concatenation ([1], rediscovered in [2] as well as in [3]):

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

*φ*((

*x*

_{1},…,

*x*

_{si})) := (

*x*

_{1},…,

*x*

_{si}, 0,…, 0).

Furthermore, choose *r* different elements *α*_{i} ∈ **F**_{b} and let *p*_{i} ∈ **F**_{b}[*X*] be defined by *p*_{i} := (*X* – *α*_{1}) ⋯ (*X* – *α*_{i−1}) for *i* = 1,…, *r*.

Then

*p*

_{i}(

*α*

_{r})

*φ*

_{i1}(

**x**_{i}),…,

*p*

_{i}(

*α*

_{r})

*φ*

_{ir}(

**x**_{i}) :

**x**_{i}∈

*C*

_{i}for

*i*= 1,…,

*r*

is an (*s*, *N*, *d*)-code over **F**_{b} with

*s*=

*s*

_{i},

*N*=

*N*

_{i}, and

*d*≥ (

*r*+ 1 –

*i*)

*d*

_{i}.

If the C_{i} are linear [*s*_{i}, *n*_{i}, *d*_{i}]-codes with *N*_{i} = *q*^{ni} and *n*_{i}×*s*_{i} generator matrices **G**_{i} for all *i* = 1,…, *r*, the resulting code is linear with generator matrix

where *φ*_{ij} is applied to the row vectors of **G**_{i}.

In the context of orthogonal arrays *r* linear OAs A_{1},…, A_{r} with parameters OA(*b*^{mi}, *s*_{i},**F**_{b}, *k*_{i}) and *s*_{1} ≤ ⋯ ≤ *s*_{r} can be used for constructing an OA(*b*^{m1+…+mr}, *s*_{1} + … + *s*_{r},**F**_{b}, *k*) with *k* = min{*rk*_{1} + (*r* – 1),(*r* – 1)*k*_{2} + (*r* – 2),…, 1*k*_{r} +0}.

### Special Cases

For

*r*= 1 this construction yields C = C_{1}.For

*r*= 2,*α*_{1}= 0, and*α*_{1}= 1 it is identical to the (*u*,*u*+*v*)-construction.For

*r*= 3, it is essentially the (*u*,*u*−*v*,*u*+*v*+*w*)-construction.

### References

[1] | È. L. Blokh and Victor V. Zyablov. Coding of generalized concatenated codes. Problems of Information Transmission, 10:218–222, 1974. |

[2] | Tim Blackmore and Graham H. Norton. Matrix-product codes over F_{q}.Applicable Algebra in Engineering, Communication and Computing, 12(6):477–500, December 2001.doi:10.1007/PL00004226 MR1873271 (2002m:94054) |

[3] | Ferruh Özbudak and Henning Stichtenoth. Note on Niederreiter-Xing’s propagation rule for linear codes. Applicable Algebra in Engineering, Communication and Computing, 13(1):53–56, April 2002.doi:10.1007/s002000100091 |

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