MinT - Best Known (67−45, 67, s)-Nets in Base 64

Best Known (67−45, 67, s)-Nets in Base 64

(67−45, 67, 177)-Net over F₆₄ — Constructive and digital

Digital (22, 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]

(67−45, 67, 288)-Net in Base 64 — Constructive

(22, 67, 288)-net in base 64, using

24 times m-reduction [i] based on (22, 91, 288)-net in base 64, using
- base change [i] based on digital (9, 78, 288)-net over F₁₂₈, using
  - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
      - a function field by SÃ©mirat [i]

(67−45, 67, 342)-Net over F₆₄ — Digital

Digital (22, 67, 342)-net over F₆₄, using

t-expansion [i] based on digital (20, 67, 342)-net over F₆₄, using
- net from sequence [i] based on digital (20, 341)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 20 and N(F) ≥ 342, using
    - a curve from the manYPoints database [i]

(67−45, 67, 37664)-Net in Base 64 — Upper bound on s

There is no (22, 67, 37665)-net in base 64, because

1 times m-reduction [i] would yield (22, 66, 37665)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 161477 655500 494055 754183 614481 706952 301580 325674 996833 447988 227915 787029 035384 851887 668455 123113 140362 777879 315850 817136 > 64⁶⁶ [i]