Journal of Combinatorial Designs, 10 (2002), 403-418.
(t,m,s)-nets were defined by Niederreiter in the context of
quasi-Monte Carlo methods of numerical integration. Niederreiter pointed out
close connections to certain combinatorial and algebraic structures.
This was made precise in the work of Lawrence, Mullen and Schmid. These authors introduce a large class of finite
combinatorial structures, which we will call ordered orthogonal arrays
OOA. These OOA contain orthogonal arrays as a subclass. (t,m,s)-nets are
equivalent to another parametric subclass of OOA.
Loosely speaking a (t,m,s)_q-net is linear if it is
vectorspace over the field GF(q) with q elements. The duality between
linear codes and linear orthogonal arrays carries over to the more general
setting of linear OOA. This has been described by Rosenbloom-Tsfasman.
Our main results are generalizations of coding-theoretic construction techniques
from Hamming space to RT-space, most notably concatenation
(equivalently: Kronecker products), the (u,u+v)-construction and the
Gilbert-Varshamov bound.
Let k=m-t denote the strength of a net.
If a linear (t,m,s)_q-net exists, where
m‹s, then a linear code [s,s-m,k+1]_q exists. From this point of view it
is a basic problem (the problem of net-embeddability) to decide when a code
[s,s-m,k+1]_q can be completed to a linear net (m-k,m,s)_q.
More generally we ask when a linear OOA with certain parameters can be embedded
in a larger OOA. We speak of a theorem of Gilbert-Varshamov type if the
existence of the larger OOA can be guaranteed whenever the parameters satisfy a
certain numerical condition. In the final section we apply our theoretical
construction techniques as well as computer-generated net embeddings of
error-correcting codes to improve upon net-parameters for nets of moderate
strength and dimension defined over small fields.
Errata:
In the tables of the printed version are wrong entries.
The entry for
q=2, k=5, m=25, s=2503, should replaced by s=2053.
q=3, k=7, m=18, s=53, should replaced by s=51.
q=4, k=3, m=3, s=6, should replaced by s=5.
q=4, k=4, m=10, s=373, should replaced by s=341.
q=4, k=13, m=34, s=98, should replaced by s=96.
q=4, k=13, m=35, s=112, should replaced by s=107.
q=4, k=13, m=36, s=128, should replaced by s=119.
q=4, k=14, m=37, s=106, should replaced by s=104.
q=4, k=13, m=38, s=119, should replaced by s=115.
q=5, k=9, m=21, s=80, should replaced by s=79.
The errors are corrcted in the attached PDF.
An up to date and more comfortable online version of the tables is the MinT data base of tms-nets.
The computer results mentioned in the paper can be found here.
Download the preprint as pdf.
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