## Construction X with Algebraic-Geometric NRT-Codes

Construction X for NRT-codes allows the construction of a new NRT-code based on two linear NRT-codes C_{1} ⊂ C_{2}, such that the dimension of C_{2} and the minimum distance of C_{1} is obtained. This is bought by increasing the length of the resulting code by the length of an additional code C_{e}, which has to be chosen depending on the parameters of C_{1} and C_{2}.

Given a linear [(*s*, *T *), *n*_{1}, *d*_{1}]-code C_{1}, which is a subcode of a linear [(*s*, *T *), *n*_{2}, *d*_{2}]-code C_{2}, as well as an additional linear [(*s*_{e}, *T *), *n*_{e}, *d*_{e}]-code C_{e}, all over the same field, a new linear [(*s* + *s*_{e}, *T *), *n*_{1} + *n*_{e}, *d*_{2} + *d*_{e}]-code can be constructed provided that *n*_{1} + *n*_{e} ≤ *n*_{2} and *d*_{2} + *d*_{e} ≤ *d*_{1}. Usually C_{e} will be chosen such that *n*_{2} = *n*_{1} + *n*_{e} and *d*_{1} = *d*_{2} + *d*_{e}, however in some situations a smaller value of *n*_{e} or *d*_{e} may also yield good results.

### Construction

Let **G**_{1}, **G**_{2}, and **G**_{e} denote the generator matrices of C_{1}, C_{2}, and C_{e}, respectively, such that the rows of **G**_{1} are a subset of the rows of **G**_{2}. Let **G**_{2}ʹ denote the *n*_{e}×*s* matrix consisting of *n*_{e} rows of **G**_{2} that are not in **G**_{1}. Then the new code is defined by the (*n*_{2} + *n*_{e})×((*s* + *s*_{e}), *T *) generator matrix

All code words formed by a non-trivial linear combination of the first *n*_{1} vectors have a weight of at least *d*_{1} because these are essentially the code words of C_{1} with *s*_{e}*T * additional 0’s appended. All other non-zero code words have a weight of at least *d*_{2} + *d*_{e} because they are built using a non-zero code word from C_{2} next to a non-zero code word from C_{e}.

### Special Cases

If C

_{1}is the [(*s*,*T*), 0,*sT*+ 1]-trivial code, which is a subcode of every linear code C_{2}, construction X reduces to juxtaposition of C_{2}and C_{e}.If C

_{1}= C_{2}, the auxiliary code C_{e}must be an [*s*_{e}, 0,*s*_{e}+ 1]-trivial code and construction X reduces to embedding C_{1}in the larger space**F**_{b}^{s+se}.

### Applications

MinT applies construction X in the following situations:

Reed-Solomon NRT-codes RS(

*T*;*n*_{1},*b*) ⊂ RS(*T*;*n*_{2},*b*) with*n*_{1}<*n*_{2}(not used)Algebraic-geometric NRT-codes AG(

*T*;*F*,*n*_{1}) ⊂ AG(*T*;*F*,*n*_{2}) with*n*_{1}<*n*_{2}Dual codes of OOAs from (

*t*,*s*)-sequence

### See Also

For

*T*= 1 the special case for linear codes is obtained

### 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. “Construction X with Algebraic-Geometric NRT-Codes.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2008-04-04.
http://mint.sbg.ac.at/desc_OConsX-Goppa.html