MinT - Best Known (228−163, 228, s)-Nets in Base 4

Best Known (228−163, 228, s)-Nets in Base 4

Digital (65, 228, 66)-net over F₄, using

t-expansion [i] based on digital (49, 228, 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, 228, 99)-net over F₄, using

t-expansion [i] based on digital (61, 228, 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, 228, 402)-net over F₄, because

3 times m-reduction [i] would yield digital (65, 225, 402)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²²⁵, 402, F₄, 160) (dual of [402, 177, 161]-code), but
  - residual code [i] would yield OA(4⁶⁵, 241, S₄, 40), but
    - the linear programming bound shows that MÂ â‰¥ 28636 313347 231972 752805 477185 503707 415442 542563 939751 037786 794056 666173 178505 529917 440000 / 20 036233 121660 447341 926213 459911 614906 079382 748507 > 4⁶⁵ [i]

There is no (65, 228, 439)-net in base 4, because

1 times m-reduction [i] would yield (65, 227, 439)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 49846 114778 228981 138667 718131 825994 462644 054358 049808 360814 212269 289251 491574 267044 849274 931493 945015 789854 285890 112042 407388 578903 862460 > 4²²⁷ [i]