MinT - Standard Lengthening for NRT-Codes

Standard Lengthening for NRT-Codes

If an [s, n, d]-code C₀ contains an [s, n−1, d + 1]-subcode C₁ (as it is the case for Reed-Solomon codes, algebraic-geometric codes and other families of codes), C₀ can be lengthened to an [s + 1, n, d + 1]-code C. If G₀ is a generator matrix of C₀ such that the first n−1 rows of G₀ are a generator matrix of C₁, a generator matrix of C is given by

G := $\displaystyle \left(\vphantom{\vec{G}_{0}\begin{array}{c} 0\\ \vdots\\ 0\\ 1\end{array}}\right.$ G₀ $\displaystyle \begin{array}{c} 0\\ \vdots\\ 0\\ 1\end{array}$ $\displaystyle \left.\vphantom{\vec{G}_{0}\begin{array}{c} 0\\ \vdots\\ 0\\ 1\end{array}}\right)$ .

The same result can be obtained by applying construction X to C₁ ⊂ C₀ with the [1, 1, 1]-code without redundancy.

For NRT-codes with depth T > 1 and n > 1 a similar construction is possible. Let C₀ denote a linear [(s, T ), n, d₀]-code. Let T ʹ := min{T , n} and let C₀ contain a chain of subcodes C_i for i = 1,…, T ʹ−1 with C_Tʹ−1 ⊂ ⋯ ⊂ C₁ ⊂ C₀, dimensions dimC_i = n−i, and minimum distances d (C_i) = d_i. Furthermore assume that G₀ denote a generator matrix of C₀ such that the first n−i rows of G form a generator matrix of C_i. Then C₀ can be lengthened to an [(s + 1, T ), n, d]-code with

d = $\displaystyle \min_{{i=0,\ldots,Tʹ−1}}^{}$ d_i + i

and generator matrix

G := $\displaystyle \left(\vphantom{\vec{G}_{0}\,\middle\vert\,\begin{array}{c} \vec{0}_{(n-T)\times T}\\ \vec{I}_{T}\end{array}}\right.$ G₀ $\displaystyle \middle$ | $\displaystyle \begin{array}{c} \vec{0}_{(n-T)\times T}\\ \vec{I}_{T}\end{array}$ $\displaystyle \left.\vphantom{\vec{G}_{0}\,\middle\vert\,\begin{array}{c} \vec{0}_{(n-T)\times T}\\ \vec{I}_{T}\end{array}}\right)$

for n ≥ T and

G := $\displaystyle \left(\vphantom{\vec{G}_{0}\,\middle\vert\,\vec{0}_{n\times(T-n)}\vec{I}_{n}}\right.$ G₀ $\displaystyle \middle$ | 0_n×(T-n)I_n $\displaystyle \left.\vphantom{\vec{G}_{0}\,\middle\vert\,\vec{0}_{n\times(T-n)}\vec{I}_{n}}\right)$

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: 2015-09-03. http://mint.sbg.ac.at/desc_OStandardLengthening.html

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