Truncation for OOAs

Every (linear) ordered orthogonal array OOA(bm, s, Sb, T , k) yields a (linear) OOA(bm−u, s – 1, Sb, T , k−u) for all u with 1 ≤ u ≤ min{T , k} [1, Corollary 5.9].

Correspondingly, every linear [(s, T ), n, d]-NRT-code yields a linear [(s−1, T ), n−T + u, d−u]-code over the same field for all u with 1 ≤ u ≤ min{T , d – 1}. If u = T it can even be shown that every (not necessarily linear) ((s, T ), N, d)-code yields an ((s−1, T ), N, d−T )-code over the same field.

Construction for OOAs

The new OOA Aʹ is obtained from A as

Aʹ = {(xi,1,…, xi,T )i=2,…, s  :  x ∈ A and x1,1 = … = x1,u = 0},

i.e., by dropping one block and all runs that do not have certain fixed elements in the first u positions of this block. If H is a generator matrix of A in the form

H = $\displaystyle \left(\vphantom{\begin{array}{ccc} \vec{0}_{(m-u)\times u} & \vec{X}_{(m-u)\times(T-u)} & \vec{H}ʹ\\ & \vec{Y}_{u\times(s,T)}\end{array}}\right.$$\displaystyle \begin{array}{ccc} \vec{0}_{(m-u)\times u} & \vec{X}_{(m-u)\times(T-u)} & \vec{H}ʹ\\ & \vec{Y}_{u\times(s,T)}\end{array}$$\displaystyle \left.\vphantom{\begin{array}{ccc} \vec{0}_{(m-u)\times u} & \vec{X}_{(m-u)\times(T-u)} & \vec{H}ʹ\\ & \vec{Y}_{u\times(s,T)}\end{array}}\right)$

(which can always be achieved using elementary row operations), the (m – u)×(s−1, T )-matrix Hʹ is a generator matrix of Aʹ. For the special case u = T the matrix H is partitioned as

H = $\displaystyle \left(\vphantom{\begin{array}{cc} \vec{0}_{(m-T)\times T} & \vec{H}ʹ\\ \vec{X}_{T\times T} & \vec{Y}_{T\times(s−1,T)}\end{array}}\right.$$\displaystyle \begin{array}{cc} \vec{0}_{(m-T)\times T} & \vec{H}ʹ\\ \vec{X}_{T\times T} & \vec{Y}_{T\times(s−1,T)}\end{array}$$\displaystyle \left.\vphantom{\begin{array}{cc} \vec{0}_{(m-T)\times T} & \vec{H}ʹ\\ \vec{X}_{T\times T} & \vec{Y}_{T\times(s−1,T)}\end{array}}\right)$.

Construction for NRT-Codes

For u < T the construction cannot be applied directly in NRT-space.

For u = T the new code Cʹ is constructed by dropping one block from all code words of C, i.e., as

Cʹ = {(xi,1,…, xi,T )i=2,…, s  :  x ∈ C}

and a generator matrix of Cʹ can be obtained by dropping an arbitrary block from a generator matrix of C.

See Also

References

[1]Rudolf Schürer.
Ordered Orthogonal Arrays and Where to Find Them.
PhD thesis, University of Salzburg, Austria, August 2006. PDF

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. “Truncation for OOAs.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2024-09-05. http://mint.sbg.ac.at/desc_OMRedS.html

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