| Introduction |
Series |
Generator matrices |
Let q be a prime-power. An Mq(s,l,m,k) is an (m,sl)-matrix with
entries in GF(q) where the columns are divided into s blocks
Bj, j=1,2,...,s of l columns each, such that the following conditions
are satisfied:
whenever k=k1+...+ks, where 0 <= kj <= l for all j, then the set of
k columns consisting of the first kj columns from each Bj is linearly
independent.
The row space of an Mq(s,l,m,k) is a linear
OOAqm-k(k,s,l,q). If l=k then a digital (m-k,m,s)-net
in base q can be constructed. For details we refer to
Mullen/Schmid and Lawrence.
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.
An Mq(s,k-1,m,k) yields an
Mq(s,k,m,k) in various ways: as k-th column of block Bj we may choose
the first column of some other block Bj'.
For more Information see:
Coding-theoretic constructions for (t,m,s)-nets and ordered orthogonal arrays
and
Construction of digital nets from BCH-Codes.
The existence of a OAqm-3(3,n,q), hence the existence of a q-ary code [n,n-m,4],
implies the existence of OOAqm-3(3,s,3,q) and so of a (m-3,m,s)-net, where s=n-1 if q=2 or q even and m=3
and s=n otherwise. See Lawrence et al.
Look at caps for codes with distance 4.
K.M.Lawrence, A combinatorial characterization of
(t,m,s)-nets in base b, J. Comb. Designs 4 (1996), 275-293.
K.M.Lawrence, A.Mahalanabis, G.L.Mullen, and
W.Ch.Schmid,
Construction of digital (t,m,s)-nets from linear codes,
in S.Cohen and H.Niederreiter, editors, Finite Fields and
Applications (Glasgow, 1995), volume 233 of Lecture Notes Series
of the
London Mathematical Society, pages 189-208.
Cambridge University Press, Cambridge, 1996.
G.L.Mullen and W.Ch.Schmid, An equivalence between
(t,m,s)-nets and strongly orthogonal hypercubes,
Journal of Combinatorial Theory A 76 (1996), 164-174.
M.Yu. Rosenbloom and M.A.Tsfasman, Codes for the m-metric,
Problems of Information Transmission 33 (1997), 145-52,
translated from: l Problemy Peredachi Informatsii 33 (1996), 55-63.
In Construction of digital nets from BCH-Codes
we found the following series:
- For r >1 an M2(22r+1,4,4r,4), the generator matrix of a
OOA24r-4(4,22r+1,4,2) equivalent to a digital binary (4r-4,4r,22r+1)-net.
-
For r >2 an M2(2r+1,4,2r+1,4), the generator matrix of a
OOA22r-3(4,2r+1,4,2) equivalent to a digital binary (2r-3,2r+1,2r+1)-net.
-
For r >2, r odd, an M2(2r-2,4,2r,4), the generator matrix of a
OOA22r-4(4,2r-2,4,2) equivalent to a digital binary (2r-4,2r,2r-2)-net.
-
For r >1 an M3(3r-1,4,2r+1,4), the generator matrix of a
OOA32r-3(4,3r-1,4,2) equivalent to a digital ternary (2r-3,2r+1,3r-1)-net.
In Families of ternary (t,m,s)-nets related to BCH-codes
we found the following series:
- For r >1 an M3((32r+1)/2,4,4r,4), the generator matrix of a
OOA34r-4(4,(32r+1)/2,4,2) equivalent to a digital ternary (4r-4,4r,(32r+1)/2)-net.
-
For r >2, r odd, an M3((3r-1)/2,4,2r,4), the generator matrix of a
OOA32r-4(4,(3r-1)/2,4,2) equivalent to a digital ternary (2r-4,2r,(3r-1)/2)-net.
Our main results in
Coding-theoretic constructions for (t,m,s)-nets and ordered orthogonal arrays
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.
In the final section of this paper 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.
In Families of nets of low and medium strength the theory of primitive BCH-codes is used to construct linear
tms-nets. Among others the following binary nets are constructed
-
(3r-5,3r+1,2r-1)2
-
(3r-5,3r+2,2r-1)2
-
(5r-8,5r,2r-1)2
-
(5r-8,5r+1,2r-1)2 for r <= 8 and
(5r-7,5r+2,2r-1)2 for all r.
In A Family of Binary (t ,m, s)-Nets of Strength 5
we construct linear (2r-3,2r+2,2r+1)2-nets for all r.
Here we give some generator matrices
(q= 2, 3, 4, 5)
obtained by machine calculation.
In the first section we deal with the case l=k. As mentioned above,
an Mq(s,k,m,k), the generator matrix of a
OOAqm-k(k,s,k,q), is equivalent to a digital q-ary (m-k,m,s)-net.
Also we saw that it is sufficient to give an Mq(s,k-1,m,k).
Table entries without a link follow from one of the series mentioned above.
Note: Some browsers do not display full matrix if it has to many columns. In that case try an other browser or download the html file and open it with your text editor.
M2(17,4,8,4)
OOA24(4,17,4,2)
(4,8,17)-net
|
M2(23,4,9,4)
OOA25(4,23,4,2)
(5,9,23)-net
|
M2(32,4,10,4)
OOA26(4,32,4,2)
(6,10,32)-net
|
M2(47,4,11,4)
OOA27(4,47,4,2)
(7,11,47)-net
|
M2(65,4,12,4)
OOA28(4,65,4,2)
(8,12,65)-net
|
M2(80,4,13,4)
OOA29(4,80,4,2)
(9,13,80)-net
|
M2(128,4,14,4)
OOA210(4,128,4,2)
(10,14,128)-net
|
M2(149,4,15,4)
OOA211(4,149,4,2)
(11,15,149)-net
|
M2(257,4,16,4)
OOA212(4,257,4,2)
(12,16,257)-net
|
M2(10,5,8,5)
OOA23(5,10,5,2)
(3,8,10)-net
|
M2(14,5,9,5)
OOA24(5,14,5,2)
(4,9,14)-net
|
M2(20,5,10,5)
OOA25(5,20,5,2)
(5,10,20)-net
|
M2(26,5,11,5)
OOA26(5,26,5,2)
(6,11,26)-net
|
M2(36,5,12,5)
OOA27(5,36,5,2)
(7,12,36)-net
|
M2(45,5,13,5)
OOA28(5,45,5,2)
(8,13,45)-net
|
M2(69,5,14,5)
OOA29(5,69,5,2)
(9,14,69)-net
|
M2(77,5,15,5)
OOA210(5,77,5,2)
(10,15,77)-net
|
M2(128,5,16,5)
OOA211(5,128,5,2)
(11,16,128)-net
|
M2(140,5,17,5)
OOA212(5,140,5,2)
(12,17,140)-net
|
M2(15,6,11,6)
OOA25(6,15,6,2)
(5,11,15)-net
|
M2(21,6,12,6)
OOA26(6,21,6,2)
(6,12,21)-net
|
M2(23,6,13,6)
OOA27(6,23,6,2)
(7,13,23)-net
|
M2(26,6,14,6)
OOA28(6,26,6,2)
(8,14,26)-net
|
M2(36,6,15,6)
OOA29(6,36,6,2)
(9,15,36)-net
|
M2(42,6,16,6)
OOA210(6,42,6,2)
(10,16,42)-net
|
M2(48,6,17,6)
OOA211(6,48,6,2)
(11,17,48)-net
|
M2(64,6,18,6)
OOA212(6,64,6,2)
(12,18,64)-net
|
M2(72,6,19,6)
OOA213(6,72,6,2)
(13,19,72)-net
|
M2(79,6,20,6)
OOA214(6,79,6,2)
(14,20,79)-net
|
M2(127,6,21,6)
OOA215(6,127,6,2)
(15,21,127)-net
|
M2(13,7,12,7)
OOA25(7,13,7,2)
(5,12,13)-net
|
M2(16,7,13,7)
OOA26(7,16,7,2)
(6,13,16)-net
|
M2(20,7,14,7)
OOA27(7,20,7,2)
(7,14,20)-net
|
M2(23,7,15,7)
OOA28(7,23,7,2)
(8,15,23)-net
|
M2(28,7,16,7)
OOA29(7,28,7,2)
(9,16,28)-net
|
M2(34,7,17,7)
OOA210(7,34,7,2)
(10,17,34)-net
|
M2(41,7,18,7)
OOA211(7,41,7,2)
(11,18,41)-net
|
M2(47,7,19,7)
OOA212(7,47,7,2)
(12,19,47)-net
|
M2(58,7,20,7)
OOA213(7,58,7,2)
(13,20,58)-net
|
M2(64,7,21,7)
OOA214(7,64,7,2)
(14,21,64)-net
|
M2(11,8,13,8)
OOA25(8,11,8,2)
(5,13,11)-net
|
M2(16,8,15,8)
OOA27(8,16,8,2)
(7,15,16)-net
|
M2(19,8,16,8)
OOA28(8,19,8,2)
(8,16,19)-net
|
M2(22,8,17,8)
OOA29(8,22,8,2)
(9,17,22)-net
|
M2(26,8,18,8)
OOA210(8,26,8,2)
(10,18,26)-net
|
M2(30,8,19,8)
OOA211(8,30,8,2)
(11,19,30)-net
|
M2(35,8,20,8)
OOA212(8,35,8,2)
(12,20,35)-net
|
M2(39,8,21,8)
OOA213(8,39,8,2)
(13,21,39)-net
|
M2(10,9,14,9)
OOA25(9,10,9,2)
(5,14,10)-net
|
M2(12,9,15,9)
OOA26(9,12,9,2)
(6,15,12)-net
|
M2(14,9,16,9)
OOA27(9,14,9,2)
(7,16,14)-net
|
M2(20,9,18,9)
OOA29(9,20,9,2)
(9,18,20)-net
|
M2(23,9,19,9)
OOA210(9,23,9,2)
(10,19,23)-net
|
M2(26,9,20,9)
OOA211(9,26,9,2)
(11,20,26)-net
|
M2(29,9,21,9)
OOA212(9,29,9,2)
(12,21,29)-net
|
M2(11,10,16,10)
OOA26(10,11,10,2)
(6,16,11)-net
|
M2(13,10,17,10)
OOA27(10,13,10,2)
(7,17,13)-net
|
M2(15,10,18,10)
OOA28(10,15,10,2)
(8,18,15)-net
|
M2(17,10,19,10)
OOA29(10,17,10,2)
(9,19,17)-net
|
M2(23,10,21,10)
OOA211(10,23,10,2)
(11,21,23)-net
|
M2(12,11,18,11)
OOA27(11,12,11,2)
(7,18,12)-net
|
M2(14,11,19,11)
OOA28(11,14,11,2)
(8,19,14)-net
|
M2(16,11,20,11)
OOA29(11,16,11,2)
(9,20,16)-net
|
M2(18,11,21,11)
OOA210(11,18,11,2)
(10,21,18)-net
|
M2(11,12,19,12)
OOA27(12,11,12,2)
(7,19,11)-net
|
M2(13,12,20,12)
OOA28(12,13,12,2)
(8,20,13)-net
|
M2(15,12,21,12)
OOA29(12,15,12,2)
(9,21,15)-net
|
M2(11,13,20,13)
OOA27(13,11,13,2)
(7,20,11)-net
|
M2(12,13,21,13)
OOA28(13,12,13,2)
(8,21,12)-net
|
M3(14,4,6,4)
OOA32(4,14,4,3)
(2,6,14)-net
|
M3(26,4,7,4)
OOA33(4,26,4,3)
(3,7,26)-net
|
M3(41,4,8,4)
OOA34(4,41,4,3)
(4,8,41)-net
|
M3(80,4,9,4)
OOA35(4,80,4,3)
(5,9,80)-net
|
M3(121,4,10,4)
OOA36(4,121,4,3)
(6,10,121)-net
|
M3(11,5,7,5)
OOA32(5,11,5,3)
(2,7,11)-net
|
M3(18,5,8,5)
OOA33(5,18,5,3)
(3,8,18)-net
|
M3(28,5,9,5)
OOA34(5,28,5,3)
(4,9,28)-net
|
M3(38,5,10,5)
OOA35(5,38,5,3)
(5,10,38)-net
|
M3(77,5,11,5)
OOA36(5,77,5,3)
(6,11,77)-net
|
M3(95,5,12,5)
OOA37(5,95,5,3)
(7,12,95)-net
|
M3(103,5,13,5)
OOA38(5,103,5,3)
(8,13,103)-net
|
M3(13,6,9,6)
OOA33(6,13,6,3)
(3,9,13)-net
|
M3(19,6,10,6)
OOA34(6,19,6,3)
(4,10,19)-net
|
M3(25,6,11,6)
OOA35(6,25,6,3)
(5,11,25)-net
|
M3(33,6,12,6)
OOA36(6,33,6,3)
(6,12,33)-net
|
M3(42,6,13,6)
OOA37(6,42,6,3)
(7,13,42)-net
|
M3(11,7,10,7)
OOA33(7,11,7,3)
(3,10,11)-net
|
M3(15,7,11,7)
OOA34(7,15,7,3)
(4,11,15)-net
|
M3(20,7,12,7)
OOA35(7,20,7,3)
(5,12,20)-net
|
M3(26,7,13,7)
OOA36(7,26,7,3)
(6,13,26)-net
|
M3(34,7,14,7)
OOA37(7,34,7,3)
(7,14,34)-net
|
M3(14,8,12,8)
OOA34(8,14,8,3)
(4,12,14)-net
|
M3(17,8,13,8)
OOA35(8,17,8,3)
(5,13,17)-net
|
M3(22,8,14,8)
OOA36(8,22,8,3)
(6,14,22)-net
|
M3(15,9,14,9)
OOA35(9,15,9,3)
(5,14,15)-net
|
M4(10,4,5,4)
OOA41(4,10,4,4)
(1,5,10)-net
|
M4(19,4,6,4)
OOA42(4,19,4,4)
(2,6,19)-net
|
M4(32,4,7,4)
OOA43(4,32,4,4)
(3,7,32)-net
|
M4(85,4,8,4)
OOA44(4,85,4,4)
(4,8,85)-net
|
M4(171,4,9,4)
OOA45(4,171,4,4)
(5,9,171)-net
|
M4(341,4,10,4)
OOA46(4,341,4,4)
(6,10,341)-net
|
M4(683,4,11,4)
OOA47(4,683,4,4)
(7,11,683)-net
|
M4(8,5,6,5)
OOA41(5,8,5,4)
(1,6,8)-net
|
M4(16,5,7,5)
OOA42(5,16,5,4)
(2,7,16)-net
|
M4(26,5,8,5)
OOA43(5,26,5,4)
(3,8,26)-net
|
M4(36,5,9,5)
OOA44(5,36,5,4)
(4,9,36)-net
|
M4(64,5,10,5)
OOA45(5,64,5,4)
(5,10,64)-net
|
M4(81,5,11,5)
OOA46(5,81,5,4)
(6,11,81)-net
|
M4(12,6,8,6)
OOA42(6,12,6,4)
(2,8,12)-net
|
M4(18,6,9,6)
OOA43(6,18,6,4)
(3,9,18)-net
|
M4(26,6,10,6)
OOA44(6,26,6,4)
(4,10,26)-net
|
M4(34,6,11,6)
OOA45(6,34,6,4)
(5,11,34)-net
|
M4(15,7,10,7)
OOA43(7,15,7,4)
(3,10,15)-net
|
M4(20,7,11,7)
OOA44(7,20,7,4)
(4,11,20)-net
|
M4(17,8,12,8)
OOA44(8,17,8,4)
(4,12,17)-net
|
M5(12,4,5,4)
OOA51(4,12,4,5)
(1,5,12)-net
|
M5(27,4,6,4)
OOA52(4,27,4,5)
(2,6,27)-net
|
M5(44,4,7,4)
OOA53(4,44,4,5)
(3,7,44)-net
|
M5(78,4,8,4)
OOA54(4,78,4,5)
(4,8,78)-net
|
M5(137,4,9,4)
OOA55(4,137,4,5)
(5,9,137)-net
|
M5(21,5,7,5)
OOA52(5,21,5,5)
(2,7,21)-net
|
M5(33,5,8,5)
OOA53(5,33,5,5)
(3,8,33)-net
|
M5(46,5,9,5)
OOA54(5,46,5,5)
(4,9,46)-net
|
M5(68,5,10,5)
OOA55(5,68,5,5)
(5,10,68)-net
|
M5(14,6,8,6)
OOA52(6,14,6,5)
(2,8,14)-net
|
M5(27,6,9,6)
OOA53(6,27,6,5)
(3,9,27)-net
|
M5(33,6,10,6)
OOA54(6,33,6,5)
(4,10,33)-net
|
M5(25,7,10,7)
OOA52(7,25,7,5)
(3,10,25)-net
|
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