MinT - Best Known (67−53, 67, s)-Nets in Base 64

Best Known (67−53, 67, s)-Nets in Base 64

(67−53, 67, 177)-Net over F₆₄ — Constructive and digital

Digital (14, 67, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 67, 177)-net over F₆₄, using
- net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
    - a function field by GarcÃa and Quoos [i]

(67−53, 67, 192)-Net in Base 64 — Constructive

(14, 67, 192)-net in base 64, using

10 times m-reduction [i] based on (14, 77, 192)-net in base 64, using
- base change [i] based on digital (3, 66, 192)-net over F₁₂₈, using
  - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
      - a function field by SÃ©mirat [i]

(67−53, 67, 257)-Net over F₆₄ — Digital

Digital (14, 67, 257)-net over F₆₄, using

t-expansion [i] based on digital (12, 67, 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]

(67−53, 67, 6426)-Net in Base 64 — Upper bound on s

There is no (14, 67, 6427)-net in base 64, because

1 times m-reduction [i] would yield (14, 66, 6427)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 161475 134645 129634 505961 896302 998494 393886 443425 806866 573751 993393 734221 831798 626024 361454 025301 232835 734622 592620 019025 > 64⁶⁶ [i]