MinT - Best Known (66, 66+122, s)-Nets in Base 3

Best Known (66, 66+122, s)-Nets in Base 3

Digital (66, 188, 48)-net over F₃, using

Digital (66, 188, 64)-net over F₃, using

t-expansion [i] based on digital (49, 188, 64)-net over F₃, using
- net from sequence [i] based on digital (49, 63)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 49 and N(F) ≥ 64, using
    - a curve from the manYPoints database [i]

There is no digital (66, 188, 249)-net over F₃, because

2 times m-reduction [i] would yield digital (66, 186, 249)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁸⁶, 249, F₃, 120) (dual of [249, 63, 121]-code), but
  - residual code [i] would yield OA(3⁶⁶, 128, S₃, 40), but
    - the linear programming bound shows that MÂ â‰¥ 1 611238 946031 791724 264759 395088 592285 928104 092842 494630 503305 571976 656701 593621 859307 702786 294582 883084 776683 289434 373704 105944 992140 198769 627517 455165 575300 524097 890798 802806 487547 361269 125679 854146 910198 196389 586965 840237 090647 747006 364719 806300 187540 016277 435514 067436 922172 504295 / 51293 380536 656250 007170 433922 291185 671132 071753 215231 290065 315554 014551 516315 231224 173693 592682 977163 227707 245881 226433 262427 420593 916503 524439 740836 602894 330584 784307 101346 736157 442225 712691 935994 375094 931626 844171 580816 568822 030521 808140 786554 738304 > 3⁶⁶ [i]

There is no (66, 188, 289)-net in base 3, because

13 times m-reduction [i] would yield (66, 175, 289)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3¹⁷⁵, 289, S₃, 109), but
  - 11 times code embedding in larger space [i] would yield OA(3¹⁸⁶, 300, S₃, 109), but
    - the linear programming bound shows that MÂ â‰¥ 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]