MinT - Best Known (63, 174, s)-Nets in Base 3

Best Known (63, 174, s)-Nets in Base 3

Digital (63, 174, 48)-net over F₃, using

Digital (63, 174, 64)-net over F₃, using

t-expansion [i] based on digital (49, 174, 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 (63, 174, 261)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹⁷⁴, 261, F₃, 111) (dual of [261, 87, 112]-code), but
- residual code [i] would yield OA(3⁶³, 149, S₃, 37), but
  - the linear programming bound shows that MÂ â‰¥ 46 631470 124673 049327 400235 264470 858493 236508 054985 173899 541299 524620 475842 712041 549727 037290 684522 127992 940560 161444 675369 235323 506654 361935 251741 549444 156200 159989 526202 879027 434375 / 37 524831 135040 905996 809144 831977 213264 438373 079442 262970 729827 862827 594743 088766 089529 538795 972448 769675 682777 166896 727335 845836 387566 113951 189899 042231 > 3⁶³ [i]

There is no (63, 174, 286)-net in base 3, because

2 times m-reduction [i] would yield (63, 172, 286)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3¹⁷², 286, S₃, 109), but
  - 14 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]