MinT - Best Known (50, 50+61, s)-Nets in Base 16

Best Known (50, 50+61, s)-Nets in Base 16

Digital (50, 111, 243)-net over F₁₆, using

t-expansion [i] based on digital (48, 111, 243)-net over F₁₆, using
- net from sequence [i] based on digital (48, 242)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 48 and N(F) ≥ 243, using
    - a function field by GarcÃa and Quoos [i]

Digital (50, 111, 255)-net over F₁₆, using

(50, 111, 257)-net in base 16, using

3 times m-reduction [i] based on (50, 114, 257)-net in base 16, using
- base change [i] based on digital (12, 76, 257)-net over F₆₄, using
  - net from sequence [i] based on digital (12, 256)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 12 and N(F) ≥ 257, using
      - K_1,2 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₆₄ [i]

There is no (50, 111, 20868)-net in base 16, because

1 times m-reduction [i] would yield (50, 110, 20868)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 843139 179217 617352 635654 447897 667384 360251 011522 687983 664165 848866 244552 958069 878233 915482 202864 561332 596677 825770 964585 113021 770976 > 16¹¹⁰ [i]