## Standard Lengthening for NRT-Codes

If an [*s*, *n*, *d*]-code C_{0} contains an [*s*, *n*−1, *d* + 1]-subcode C_{1} (as it is the case for Reed-Solomon codes, algebraic-geometric codes and other families of codes), C_{0} can be lengthened to an [*s* + 1, *n*, *d* + 1]-code C. If **G**_{0} is a generator matrix of C_{0} such that the first *n*−1 rows of **G**_{0} are a generator matrix of C_{1}, a generator matrix of C is given by

*:=*

**G**

**G**_{0}.

The same result can be obtained by applying construction X to C_{1} ⊂ C_{0} with the [1, 1, 1]-code without redundancy.

For NRT-codes with depth *T * > 1 and *n* > 1 a similar construction is possible. Let C_{0} denote a linear [(*s*, *T *), *n*, *d*_{0}]-code. Let *T *ʹ := min{*T *, *n*} and let C_{0} contain a chain of subcodes C_{i} for *i* = 1,…, *T *ʹ−1 with C_{Tʹ−1} ⊂ ⋯ ⊂ C_{1} ⊂ C_{0}, dimensions dimC_{i} = *n*−*i*, and minimum distances *d* (C_{i}) = *d*_{i}. Furthermore assume that **G**_{0} denote a generator matrix of C_{0} such that the first *n*−*i* rows of * G* form a generator matrix of C

_{i}. Then C

_{0}can be lengthened to an [(

*s*+ 1,

*T*),

*n*,

*d*]-code with

*d*=

*d*

_{i}+

*i*

and generator matrix

*:=*

**G**

**G**_{0}|

for *n* ≥ *T * and

*:=*

**G**

**G**_{0}| 0

_{n×(T-n)}

**I**_{n}

for *n* ≤ *T *.

Note that this result cannot be obtained using construction X in a straightforward way! It is only possible because the extension code C_{e} defined by the generator matrix **I**_{Tʹ} is an unequal error protection (UEP) NRT-code: If D_{i} denotes the code defined by the first *i* rows of **I**_{Tʹ}, then *d* (C_{e} ∖ D_{i}) = *i* + 1 for *i* = 0,…, *T *ʹ−1 (cf. [1]).

### References

[1] | Jürgen Bierbrauer, Yves Edel, and Ludo Tolhuizen. New codes via the lengthening of BCH codes with UEP codes. Finite Fields and Their Applications, 5(4):345–353, October 1999.doi:10.1006/ffta.1999.0245 MR1711841 (2000j:94027) |

### 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. “Standard Lengthening for NRT-Codes.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2008-04-04.
http://mint.sbg.ac.at/desc_OStandardLengthening.html