Generalized (uu+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 C1,…, Cr be r (s1, N1, d1)-codes, all over Fb, with br and s1 ≤ ⋯ ≤ sr. Let φij : FbsiFbsj for ij denote the embedding of Fbsi in Fbsj defined by

φ((x1,…, xsi)) := (x1,…, xsi, 0,…, 0).

Furthermore, choose r different elements αiFb and let piFb[X] be defined by pi := (Xα1) ⋯ (Xαi−1) for i = 1,…, r.

Then

C := $\displaystyle \left\{\vphantom{ \left(\sum_{i=1}^{1}p_{i}(\alpha_{r})\varphi_{i… …(\vec{x}_{i})\right) : \vec{x}_{i}\in C_{i}\textrm{ for }i=1,\ldots,r}\right.$$\displaystyle \left(\vphantom{\sum_{i=1}^{1}p_{i}(\alpha_{r})\varphi_{i1}(\vec{x}_{i}),\ldots,\sum_{i=1}^{r}p_{i}(\alpha_{r})\varphi_{ir}(\vec{x}_{i})}\right.$$\displaystyle \sum_{{i=1}}^{{1}}$pi(αr)φi1(xi),…,$\displaystyle \sum_{{i=1}}^{{r}}$pi(αr)φir(xi)$\displaystyle \left.\vphantom{\sum_{i=1}^{1}p_{i}(\alpha_{r})\varphi_{i1}(\vec{x}_{i}),\ldots,\sum_{i=1}^{r}p_{i}(\alpha_{r})\varphi_{ir}(\vec{x}_{i})}\right)$  :  xiCifori = 1,…, r$\displaystyle \left.\vphantom{ \left(\sum_{i=1}^{1}p_{i}(\alpha_{r})\varphi_{i1… …\vec{x}_{i})\right) : \vec{x}_{i}\in C_{i}\textrm{ for }i=1,\ldots,r}\right\}$

is an (s, N, d)-code over Fb with

s = $\displaystyle \sum_{{i=1}}^{{r}}$si,        N = $\displaystyle \prod_{{i=1}}^{{r}}$Ni,    and    d$\displaystyle \min_{{i=1,\ldots,r}}^{}$(r + 1 – i)di.

If the Ci are linear [si, ni, di]-codes with Ni = qni and ni×si generator matrices Gi for all i = 1,…, r, the resulting code is linear with generator matrix

$\displaystyle \left(\vphantom{\begin{array}{cccc} p_{1}(\alpha_{1})\varphi_{11}… …vdots\\ 0 & & & p_{r}(\alpha_{r})\varphi_{rr}(\vec{G}_{r})\end{array}}\right.$$\displaystyle \begin{array}{cccc} p_{1}(\alpha_{1})\varphi_{11}(\vec{G}_{1}) & … …dots & \vdots\\ 0 & & & p_{r}(\alpha_{r})\varphi_{rr}(\vec{G}_{r})\end{array}$$\displaystyle \left.\vphantom{\begin{array}{cccc} p_{1}(\alpha_{1})\varphi_{11}… …vdots\\ 0 & & & p_{r}(\alpha_{r})\varphi_{rr}(\vec{G}_{r})\end{array}}\right)$

where φij is applied to the row vectors of Gi.

In the context of orthogonal arrays r linear OAs A1,…, Ar with parameters OA(bmi, si,Fb, ki) and s1 ≤ ⋯ ≤ sr can be used for constructing an OA(bm1+…+mr, s1 + … + sr,Fb, k) with k = min{rk1 + (r – 1),(r – 1)k2 + (r – 2),…, 1kr +0}.

Special Cases

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 Fq.
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 (uu+v)-Construction.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2008-04-04. http://mint.sbg.ac.at/desc_CGeneralizedUUPlusV.html

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