(u, u+v)-Construction

Given two (linear) orthogonal arrays OA(Mi, si, Sb, ki) with s1 ≤ s2, a (linear) OA(M1M2, s1 + s2, Sb, k) with k = min{2k1 +1, k2} can be constructed.

Correspondingly, given two (linear) (si, Ni, di)-codes over the same field with s1 ≤ s2, a (linear) (s1 + s2, N1N2, d) with d = min{2d1, d2} can be constructed.

The Construction for Orthogonal Arrays

Let A1 and A2 denote the orthogonal arrays with parameters OA(M1, s1, Sb, k1) and OA(M2, s2, Sb, k2), respectively, and let s1 ≤ s2. Then the resulting orthogonal array A with parameters OA(M1M2, s1 + s2, Sb, k) is given by

A = {(u – π(v),v)  :  u ∈ A1,v ∈ A2}

where π : Sbs2→Sbs1 is the projection selecting the first s1 coordinates.

If A1 and A2 are linear with Mi = bmi and mi×si generator matrices Hi, the generator matrix of A is given by

$\displaystyle \left(\vphantom{\begin{array}{cc} \vec{H}_{1} & \vec{0}_{m_{1}\times s_{2}}\\ -\pi(\vec{H}_{2}) & \vec{H}_{2}\end{array}}\right.$$\displaystyle \begin{array}{cc} \vec{H}_{1} & \vec{0}_{m_{1}\times s_{2}}\\ -\pi(\vec{H}_{2}) & \vec{H}_{2}\end{array}$$\displaystyle \left.\vphantom{\begin{array}{cc} \vec{H}_{1} & \vec{0}_{m_{1}\times s_{2}}\\ -\pi(\vec{H}_{2}) & \vec{H}_{2}\end{array}}\right)$

with π(H2) denoting the first s1 columns of H2.

The Construction for Codes

Let C1 be an (s1, N1, d1)-code and let C2 be an (s2, N2, d2)-code, both over Fb with s1 ≤ s2. Then it is shown in [1] that the set of vectors

C = {(u,(u, 0n1×(s2-s1)) + v)  :  u ∈ C1,v ∈ C2}

is an (s1 + s2, N1N2, d)-code over Fb with d = min{2d1, d2}. The result for s1 = s2 can already be found in [2].

If C1 and C2 are linear with Ni = bni, mi = si – ni, ni×s generator matrices Gi, and mi×si parity check matrices Hi, the generator matrix of the new linear [s1 + s2, n1 + n2, d]-code C is given by

$\displaystyle \left(\vphantom{\begin{array}{cc} \vec{G}_{1} & \vec{G}_{1},\vec{… …(s_{2}-s_{1})}\\ \vec{0}_{n_{2}\times s_{1}} & \vec{G}_{2}\end{array}}\right.$$\displaystyle \begin{array}{cc} \vec{G}_{1} & \vec{G}_{1},\vec{0}_{n_{1}\times(s_{2}-s_{1})}\\ \vec{0}_{n_{2}\times s_{1}} & \vec{G}_{2}\end{array}$$\displaystyle \left.\vphantom{\begin{array}{cc} \vec{G}_{1} & \vec{G}_{1},\vec{… …(s_{2}-s_{1})}\\ \vec{0}_{n_{2}\times s_{1}} & \vec{G}_{2}\end{array}}\right)$,

its parity check matrix is shown above.

See Also

References

[1]Neil J. A. Sloane and D. S. Whitehead.
A new family of single-error correcting codes.
IEEE Transactions on Information Theory, 16(6):717–719, November 1970.
[2]Morris Plotkin.
Binary codes with specified minimum distance.
IEEE Transactions on Information Theory, 6(4):445–450, September 1960.
[3]F. Jessie MacWilliams and Neil J. A. Sloane.
The Theory of Error-Correcting Codes.
North-Holland, Amsterdam, 1977.
[4]Jacobus H. van Lint.
Introduction to Coding Theory, volume 86 of Graduate Texts in Mathematics.
Springer-Verlag, second edition, 1991.
[5]A. S. Hedayat, Neil J. A. Sloane, and John Stufken.
Orthogonal Arrays.
Springer Series in Statistics. Springer-Verlag, 1999.
[6]Jürgen Bierbrauer.
Introduction to Coding Theory.
Discrete Mathematics and its Applications. Chapman & Hall/CRC, Boca Raton, London, New York, Washington D.C., 2004.
MR2079734 (2005f:94001)

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)-Construction.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03. http://mint.sbg.ac.at/desc_CUUPlusV.html

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