MinT - Best Known (65, 65+165, s)-Nets in Base 4

Best Known (65, 65+165, s)-Nets in Base 4

Digital (65, 230, 66)-net over F₄, using

t-expansion [i] based on digital (49, 230, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

Digital (65, 230, 99)-net over F₄, using

t-expansion [i] based on digital (61, 230, 99)-net over F₄, using
- net from sequence [i] based on digital (61, 98)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 61 and N(F) ≥ 99, using
    - Drinfeld modules of rank 1 [i]

There is no digital (65, 230, 387)-net over F₄, because

1 times m-reduction [i] would yield digital (65, 229, 387)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²²⁹, 387, F₄, 164) (dual of [387, 158, 165]-code), but
  - residual code [i] would yield OA(4⁶⁵, 222, S₄, 41), but
    - the linear programming bound shows that MÂ â‰¥ 32 342197 523814 651165 392954 045591 141352 749935 799714 032461 858391 219530 982071 151362 048000 / 23186 656803 327364 360991 616330 320246 466025 092007 > 4⁶⁵ [i]

There is no (65, 230, 437)-net in base 4, because

1 times m-reduction [i] would yield (65, 229, 437)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 747501 651698 145615 683400 643757 719088 248991 136622 455012 003183 532071 666863 778279 001300 810739 184899 816556 943232 210849 340200 838072 585423 087100 > 4²²⁹ [i]