| 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'.

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.

## Series and constructions

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.

## 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.

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.

#### 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

| home |