MinT - Best Known (65, 224, s)-Nets in Base 4

Best Known (65, 224, s)-Nets in Base 4

Digital (65, 224, 66)-net over F₄, using

t-expansion [i] based on digital (49, 224, 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, 224, 99)-net over F₄, using

t-expansion [i] based on digital (61, 224, 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, 224, 434)-net over F₄, because

3 times m-reduction [i] would yield digital (65, 221, 434)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²²¹, 434, F₄, 156) (dual of [434, 213, 157]-code), but
  - residual code [i] would yield OA(4⁶⁵, 277, S₄, 39), but
    - the linear programming bound shows that MÂ â‰¥ 114860 870016 374579 218653 717961 476899 219304 784975 141224 743527 918706 993971 303087 603712 / 81 574206 852274 374488 234675 560995 174637 976327 > 4⁶⁵ [i]

There is no (65, 224, 443)-net in base 4, because

1 times m-reduction [i] would yield (65, 223, 443)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 210 006658 102068 694722 896576 026700 800512 758279 467955 854575 088915 250143 743954 388686 150371 047806 772883 426997 386301 956341 861256 274434 714560 > 4²²³ [i]