MinT - Best Known (57−35, 57, s)-Nets in Base 64

Best Known (57−35, 57, s)-Nets in Base 64

(57−35, 57, 193)-Net over F₆₄ — Constructive and digital

Digital (22, 57, 193)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 17, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
      - the rational function field F₆₄(x) [i]
    - Niederreiter sequence [i]
2. digital (5, 40, 128)-net over F₆₄, using
  - net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
      - a function field by SÃ©mirat [i]

(57−35, 57, 288)-Net in Base 64 — Constructive

(22, 57, 288)-net in base 64, using

34 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]

(57−35, 57, 342)-Net over F₆₄ — Digital

Digital (22, 57, 342)-net over F₆₄, using

t-expansion [i] based on digital (20, 57, 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]

(57−35, 57, 101470)-Net in Base 64 — Upper bound on s

There is no (22, 57, 101471)-net in base 64, because

1 times m-reduction [i] would yield (22, 56, 101471)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 140002 402708 652647 686814 873251 636047 983999 810550 964951 356025 059447 751888 549798 974450 668642 945667 333566 > 64⁵⁶ [i]