MinT - Generalized (u, u+v)-Construction

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 C₁,…, C_r be r (s_i, N_i, d_i)-codes, all over F_b, with b ≥ r and s₁ ≤ ⋯ ≤ s_r. Let φ_ij : F_b^s_i→F_b^s_j for i ≤ j denote the embedding of F_b^s_i in F_b^s_j defined by

φ((x₁,…, x_{s_i})) := (x₁,…, x_{s_i}, 0,…, 0).

Furthermore, choose r different elements α_i ∈ F_b and let p_i ∈ F_b[X] be defined by p_i := (X – α₁) ⋯ (X – α_i−1) for i = 1,…, r.

Then

C := $\displaystyle \left\{\vphantom{ \left(\sum_{i=1}^{1}p_{i}(\alpha_{r})\varphi_{i… …(\vec{x}_{i})\right)\,:\,\vec{x}_{i}\in C_{i}\textrm{ for }i=1,\ldots,r}\right.$ $\displaystyle \left(\vphantom{\sum_{i=1}^{1}p_{i}(\alpha_{r})\varphi_{i1}(\vec{x}_{i}),\ldots,\sum_{i=1}^{r}p_{i}(\alpha_{r})\varphi_{ir}(\vec{x}_{i})}\right.$ $\displaystyle \sum_{{i=1}}^{{1}}$ p_i(α_r)φ_i1(x_i),…, $\displaystyle \sum_{{i=1}}^{{r}}$ p_i(α_r)φ_ir(x_i) $\displaystyle \left.\vphantom{\sum_{i=1}^{1}p_{i}(\alpha_{r})\varphi_{i1}(\vec{x}_{i}),\ldots,\sum_{i=1}^{r}p_{i}(\alpha_{r})\varphi_{ir}(\vec{x}_{i})}\right)$ : x_i ∈ C_ifori = 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 F_b with

s = $\displaystyle \sum_{{i=1}}^{{r}}$ s_i, N = $\displaystyle \prod_{{i=1}}^{{r}}$ N_i, and d ≥ $\displaystyle \min_{{i=1,\ldots,r}}^{}$ (r + 1 – i)d_i.

If the C_i are linear [s_i, n_i, d_i]-codes with N_i = q^n_i and n_i×s_i generator matrices G_i 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 φ_ij is applied to the row vectors of G_i.

In the context of orthogonal arrays r linear OAs A₁,…, A_r with parameters OA(b^m_i, s_i,F_b, k_i) and s₁ ≤ ⋯ ≤ s_r can be used for constructing an OA(b^m₁+…+m_r, s₁ + … + s_r,F_b, k) with k = min{rk₁ + (r – 1),(r – 1)k₂ + (r – 2),…, 1k_r +0}.

Special Cases

For r = 1 this construction yields C = C₁.
For r = 2, α₁ = 0, and α₁ = 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

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