MinT - Best Known (66, 66+116, s)-Nets in Base 3

Best Known (66, 66+116, s)-Nets in Base 3

Digital (66, 182, 48)-net over F₃, using

Digital (66, 182, 64)-net over F₃, using

t-expansion [i] based on digital (49, 182, 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, 182, 279)-net over F₃, because

2 times m-reduction [i] would yield digital (66, 180, 279)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁸⁰, 279, F₃, 114) (dual of [279, 99, 115]-code), but
  - residual code [i] would yield OA(3⁶⁶, 164, S₃, 38), but
    - the linear programming bound shows that MÂ â‰¥ 28 879890 598693 981767 858904 700078 804544 010519 089400 319761 254528 195493 544126 197753 947774 003837 783343 880912 553452 712668 486144 993991 575869 777324 277760 / 903400 545294 735183 153424 590171 611234 670868 730655 231672 024594 670239 604239 180914 860835 373684 382446 610001 352164 878597 > 3⁶⁶ [i]

There is no (66, 182, 289)-net in base 3, because

7 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]