MinT - Best Known (54, 54+97, s)-Nets in Base 3

Best Known (54, 54+97, s)-Nets in Base 3

Digital (54, 151, 48)-net over F₃, using

Digital (54, 151, 64)-net over F₃, using

t-expansion [i] based on digital (49, 151, 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 (54, 151, 223)-net over F₃, because

1 times m-reduction [i] would yield digital (54, 150, 223)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁵⁰, 223, F₃, 96) (dual of [223, 73, 97]-code), but
  - residual code [i] would yield OA(3⁵⁴, 126, S₃, 32), but
    - the linear programming bound shows that MÂ â‰¥ 15 551175 343489 816312 839781 958526 754046 650025 716899 066271 597025 177434 186311 225436 494080 388171 855211 042571 528095 306878 344834 964322 752687 664073 776320 / 247669 228492 701837 800287 038860 817164 439353 182480 938178 042680 577677 782903 404567 720365 536303 663330 806259 025721 550363 325847 > 3⁵⁴ [i]

There is no (54, 151, 246)-net in base 3, because

1 times m-reduction [i] would yield (54, 150, 246)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 387451 354784 573788 943541 692471 637023 518887 848993 878114 282616 648659 475937 > 3¹⁵⁰ [i]