Best Known (189−121, 189, s)-Nets in Base 3
(189−121, 189, 48)-Net over F3 — Constructive and digital
Digital (68, 189, 48)-net over F3, using
- t-expansion [i] based on digital (45, 189, 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
(189−121, 189, 72)-Net over F3 — Digital
Digital (68, 189, 72)-net over F3, using
- t-expansion [i] based on digital (67, 189, 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
- net from sequence [i] based on digital (67, 71)-sequence over F3, using
(189−121, 189, 274)-Net over F3 — Upper bound on s (digital)
There is no digital (68, 189, 275)-net over F3, because
- 1 times m-reduction [i] would yield digital (68, 188, 275)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3188, 275, F3, 120) (dual of [275, 87, 121]-code), but
- residual code [i] would yield OA(368, 154, S3, 40), but
- the linear programming bound shows that M ≥ 1543 058592 964362 627616 920522 828367 173876 303246 223002 800564 908980 622587 014726 821210 979933 875704 730374 337724 405572 586571 638574 502256 542100 430421 331132 939258 230113 015935 536697 467246 600185 034473 231953 255137 438105 864201 753265 543937 797738 079087 583095 / 5 223863 926286 388649 771330 591111 717512 853140 916572 381721 122477 739536 180288 144562 059465 186434 644433 474885 418173 813040 846526 036560 913385 399148 362432 124880 619677 394872 674126 851680 403717 510145 188107 930768 075751 656943 > 368 [i]
- residual code [i] would yield OA(368, 154, S3, 40), but
- extracting embedded orthogonal array [i] would yield linear OA(3188, 275, F3, 120) (dual of [275, 87, 121]-code), but
(189−121, 189, 290)-Net in Base 3 — Upper bound on s
There is no (68, 189, 291)-net in base 3, because
- 12 times m-reduction [i] would yield (68, 177, 291)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3177, 291, S3, 109), but
- 9 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]
- 9 times code embedding in larger space [i] would yield OA(3186, 300, S3, 109), but
- extracting embedded orthogonal array [i] would yield OA(3177, 291, S3, 109), but