Best Known (190−123, 190, s)-Nets in Base 3
(190−123, 190, 48)-Net over F3 — Constructive and digital
Digital (67, 190, 48)-net over F3, using
- t-expansion [i] based on digital (45, 190, 48)-net over F3, using
- net from sequence [i] based on digital (45, 47)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 45 and N(F) ≥ 48, using
- net from sequence [i] based on digital (45, 47)-sequence over F3, using
(190−123, 190, 72)-Net over F3 — Digital
Digital (67, 190, 72)-net over F3, using
- net from sequence [i] based on digital (67, 71)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 67 and N(F) ≥ 72, using
(190−123, 190, 246)-Net over F3 — Upper bound on s (digital)
There is no digital (67, 190, 247)-net over F3, because
- extracting embedded orthogonal array [i] would yield linear OA(3190, 247, F3, 123) (dual of [247, 57, 124]-code), but
- residual code [i] would yield OA(367, 123, S3, 41), but
- the linear programming bound shows that M ≥ 95701 386129 167166 923308 851077 729205 755247 753256 994296 019812 866271 686259 905678 086747 373964 096192 633285 618288 594470 403818 339365 270251 414448 366241 045444 832354 852952 005712 116408 640186 608601 069925 134925 254443 011444 426759 / 952 500022 925140 246902 196538 812302 729173 238207 945067 906882 779057 896453 751466 708206 167368 761900 485092 912190 846196 557708 634886 092699 723386 777232 538945 438378 302685 186019 204098 620520 999835 > 367 [i]
- residual code [i] would yield OA(367, 123, S3, 41), but
(190−123, 190, 289)-Net in Base 3 — Upper bound on s
There is no (67, 190, 290)-net in base 3, because
- 14 times m-reduction [i] would yield (67, 176, 290)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3176, 290, S3, 109), but
- 10 times code embedding in larger space [i] would yield OA(3186, 300, S3, 109), but
- the linear programming bound shows that M ≥ 744 646766 850603 526568 601268 797422 932026 535817 992617 489413 629925 481692 835741 945421 210090 969950 381274 252816 504020 353660 158683 457200 925516 493661 440389 831805 338854 508258 811020 990136 433923 762970 235494 388055 889637 880458 633459 997426 817105 857222 614437 226587 / 10512 885684 310613 723066 595609 767913 116130 406095 478720 857818 594175 294259 025889 504412 156455 324830 169263 105843 888569 812652 562285 873819 443701 107860 316189 843301 339995 > 3186 [i]
- 10 times code embedding in larger space [i] would yield OA(3186, 300, S3, 109), but
- extracting embedded orthogonal array [i] would yield OA(3176, 290, S3, 109), but