Best Known (216−140, 216, s)-Nets in Base 3
(216−140, 216, 51)-Net over F3 — Constructive and digital
Digital (76, 216, 51)-net over F3, using
- net from sequence [i] based on digital (76, 50)-sequence over F3, using
- base reduction for sequences [i] based on digital (13, 50)-sequence over F9, using
- s-reduction based on digital (13, 63)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 13 and N(F) ≥ 64, using
- s-reduction based on digital (13, 63)-sequence over F9, using
- base reduction for sequences [i] based on digital (13, 50)-sequence over F9, using
(216−140, 216, 84)-Net over F3 — Digital
Digital (76, 216, 84)-net over F3, using
- t-expansion [i] based on digital (71, 216, 84)-net over F3, using
- net from sequence [i] based on digital (71, 83)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 71 and N(F) ≥ 84, using
- net from sequence [i] based on digital (71, 83)-sequence over F3, using
(216−140, 216, 279)-Net over F3 — Upper bound on s (digital)
There is no digital (76, 216, 280)-net over F3, because
- 2 times m-reduction [i] would yield digital (76, 214, 280)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3214, 280, F3, 138) (dual of [280, 66, 139]-code), but
- residual code [i] would yield OA(376, 141, S3, 46), but
- the linear programming bound shows that M ≥ 1239 425902 701167 359387 634027 779408 519762 709262 383373 969093 013055 394189 451321 855434 419134 955995 578843 428108 571632 231710 187419 868518 256583 326241 168944 252882 398735 979398 922721 553289 167661 439278 006937 289846 459956 997182 334962 086820 293756 300752 234920 627744 965167 159378 951150 671742 092706 567678 330701 / 619 151021 745863 958229 474937 131529 581480 994844 744514 155471 198189 416920 526044 642490 200443 004626 495589 553207 277337 934597 756407 331144 456306 914384 534916 851146 560682 812459 331622 620796 601820 969552 710112 741666 982370 750873 449648 332022 999009 114022 310456 252406 916425 > 376 [i]
- residual code [i] would yield OA(376, 141, S3, 46), but
- extracting embedded orthogonal array [i] would yield linear OA(3214, 280, F3, 138) (dual of [280, 66, 139]-code), but
(216−140, 216, 298)-Net in Base 3 — Upper bound on s
There is no (76, 216, 299)-net in base 3, because
- 31 times m-reduction [i] would yield (76, 185, 299)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3185, 299, S3, 109), but
- 1 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]
- 1 times code embedding in larger space [i] would yield OA(3186, 300, S3, 109), but
- extracting embedded orthogonal array [i] would yield OA(3185, 299, S3, 109), but