MinT - Best Known (96−51, 96, s)-Nets in Base 16

Best Known (96−51, 96, s)-Nets in Base 16

Digital (45, 96, 225)-net over F₁₆, using

t-expansion [i] based on digital (40, 96, 225)-net over F₁₆, using
- net from sequence [i] based on digital (40, 224)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 40 and N(F) ≥ 225, using
    - a function field by GarcÃa and Quoos [i]

Digital (45, 96, 256)-net over F₁₆, using

(45, 96, 257)-net in base 16, using

3 times m-reduction [i] based on (45, 99, 257)-net in base 16, using
- base change [i] based on digital (12, 66, 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 (45, 96, 25525)-net in base 16, because

1 times m-reduction [i] would yield (45, 95, 25525)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 463715 195031 270129 285720 857843 914370 750341 068134 117758 420287 635437 055570 513957 549963 424572 499143 675779 567027 759376 > 16⁹⁵ [i]