(u, u−v, u+v+w)-Construction

Let Ci be (si, Ni, di)-codes over Fb for i = 1, 2, 3 with s1 ≤ s2 ≤ s3 and b ≥ 3. For i ≤ j let φi,j : Fbsi→Fbsj denote the embedding of Fbsi in Fbsj defined by

φ((x1,…, xsi)) := (x1,…, ssi, 0,…, 0)

and let α ∈ Fb ∖ {0, 1}. Then the set of vectors

C = {(u, φ1,2(u) + v, φ1,3(u) + αφ2,3(v) + w)  :  u ∈ C1,v ∈ C2,w ∈ C3}

is an (s1 + s2 + s3, N1N2N3, d)-code over Fb with d = min{3d1, 2d2, d3}. If the Ci are linear with Ni = bni, ni×si generator matrices Gi, mi := si – ni, and mi×si parity check matrices Hi, the generator matrix G of the new linear [s1 + s2 + s3, n1 + n2 + n3, d]-code C is given by

G = $\displaystyle \left(\vphantom{\begin{array}{ccc} \vec{G}_{1} & \varphi_{1,2}(\v… …{3}\times s_{1}} & \vec{0}_{n_{3}\times s_{2}} & \vec{G}_{3}\end{array}}\right.$$\displaystyle \begin{array}{ccc} \vec{G}_{1} & \varphi_{1,2}(\vec{G}_{1}) & \va… …c{0}_{n_{3}\times s_{1}} & \vec{0}_{n_{3}\times s_{2}} & \vec{G}_{3}\end{array}$$\displaystyle \left.\vphantom{\begin{array}{ccc} \vec{G}_{1} & \varphi_{1,2}(\v… …{3}\times s_{1}} & \vec{0}_{n_{3}\times s_{2}} & \vec{G}_{3}\end{array}}\right)$,

its parity check matrix H by

H = $\displaystyle \left(\vphantom{\begin{array}{ccc} \vec{H}_{1} & \vec{0}_{m_{1}\t… …}(\vec{H}_{3}) & -\alpha\pi_{3,2}(\vec{H}_{3}) & \vec{H}_{3}\end{array}}\right.$$\displaystyle \begin{array}{ccc} \vec{H}_{1} & \vec{0}_{m_{1}\times s_{2}} & \v… …\pi_{3,1}(\vec{H}_{3}) & -\alpha\pi_{3,2}(\vec{H}_{3}) & \vec{H}_{3}\end{array}$$\displaystyle \left.\vphantom{\begin{array}{ccc} \vec{H}_{1} & \vec{0}_{m_{1}\t… …}(\vec{H}_{3}) & -\alpha\pi_{3,2}(\vec{H}_{3}) & \vec{H}_{3}\end{array}}\right)$

with φi,j(Gi) denoting the ni×sj-matrix (Gi 0ni×(sj-si)) and πj,i(Hj) denoting the first si columns from Hj.

In the context of orthogonal arrays three linear OAs A1, A2, and A3 with parameters OA(bmi, si,Fb, ki) and with generator matrices Hi and s1 ≤ s2 ≤ s3 can be used for constructing an OA(bm1+m2+m3, s1 + s2 + s3,Fb, k) with k = min{3k1 +2, 2k2 +1, k3} and generator matrix H, where H is the parity check matrix shown above.

If b ≥ 4, this result can be generalized such that more than three codes are used, leading to the generalized (u, u + v)-construction. The special case where C1 is the trivial [0, 0]-code is identical to the (u, u + v)-construction.

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. “(u, u−v, u+v+w)-Construction.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03. http://mint.sbg.ac.at/desc_CUUMinusVUPlusVPlusW.html

Show usage of this method