MinT - Best Known (50, 50+92, s)-Nets in Base 3

Best Known (50, 50+92, s)-Nets in Base 3

Digital (50, 142, 48)-net over F₃, using

Digital (50, 142, 64)-net over F₃, using

t-expansion [i] based on digital (49, 142, 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 (50, 142, 204)-net over F₃, because

2 times m-reduction [i] would yield digital (50, 140, 204)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁴⁰, 204, F₃, 90) (dual of [204, 64, 91]-code), but
  - residual code [i] would yield OA(3⁵⁰, 113, S₃, 30), but
    - the linear programming bound shows that MÂ â‰¥ 596 570195 583823 721692 911689 409545 159333 416347 007509 353942 840802 048696 218909 803001 021000 425338 904291 291412 200954 459797 559193 112998 126796 056497 666716 210000 / 790 509540 724661 767547 670804 913087 923304 064947 691066 061183 661201 682713 479390 203130 739891 928552 692939 503065 752658 282066 469332 139629 > 3⁵⁰ [i]

There is no (50, 142, 225)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 58 464250 415798 945832 774625 980212 034924 501111 628808 405886 467442 033849 > 3¹⁴² [i]