Best Known (67, 186, s)-Nets in Base 3
(67, 186, 48)-Net over F3 — Constructive and digital
Digital (67, 186, 48)-net over F3, using
- t-expansion [i] based on digital (45, 186, 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
(67, 186, 72)-Net over F3 — Digital
Digital (67, 186, 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
(67, 186, 278)-Net over F3 — Upper bound on s (digital)
There is no digital (67, 186, 279)-net over F3, because
- 2 times m-reduction [i] would yield digital (67, 184, 279)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3184, 279, F3, 117) (dual of [279, 95, 118]-code), but
- residual code [i] would yield OA(367, 161, S3, 39), but
- the linear programming bound shows that M ≥ 7869 930401 701596 807600 626345 198645 987250 178540 123516 450258 613760 580612 231642 212759 197575 489472 106615 287923 905883 555477 227637 750454 750095 263289 921478 546175 064686 283056 062869 623904 743004 569600 / 82 148594 495356 629216 229580 554269 756505 456638 376053 336365 080668 371924 625544 809288 130216 022035 376144 707077 418285 944925 300421 885924 252263 109749 469346 381267 111193 > 367 [i]
- residual code [i] would yield OA(367, 161, S3, 39), but
- extracting embedded orthogonal array [i] would yield linear OA(3184, 279, F3, 117) (dual of [279, 95, 118]-code), but
(67, 186, 289)-Net in Base 3 — Upper bound on s
There is no (67, 186, 290)-net in base 3, because
- 10 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