MinT - Best Known (64, 223, s)-Nets in Base 4

Best Known (64, 223, s)-Nets in Base 4

Digital (64, 223, 66)-net over F₄, using

t-expansion [i] based on digital (49, 223, 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 (64, 223, 99)-net over F₄, using

t-expansion [i] based on digital (61, 223, 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 (64, 223, 410)-net over F₄, because

3 times m-reduction [i] would yield digital (64, 220, 410)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²²⁰, 410, F₄, 156) (dual of [410, 190, 157]-code), but
  - residual code [i] would yield OA(4⁶⁴, 253, S₄, 39), but
    - the linear programming bound shows that MÂ â‰¥ 87 990315 499933 092747 998978 965173 866430 179588 697159 392509 031688 470188 023139 532800 / 245601 782466 451850 219515 982198 351916 865117 > 4⁶⁴ [i]

There is no (64, 223, 434)-net in base 4, because

1 times m-reduction [i] would yield (64, 222, 434)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 50 359105 714682 208957 877705 478943 497998 698922 083759 795142 372090 708552 487125 308803 947642 735200 247209 675439 720693 299473 906559 484766 739760 > 4²²² [i]