| Introduction | Series | Generator matrices |

Introduction

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.



Series and constructions

In Construction of digital nets from BCH-Codes we found the following series: In Families of ternary (t,m,s)-nets related to BCH-codes we found the following series: 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

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.


Generator matrices of some linear OOA

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.

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Generator matrices of some linear binary OOA

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


Generator matrices of some linear ternary OOA.

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


Generator matrices of some linear quaternary OOA.

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


Generator matrices of some linear OOA for q=5.

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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