MinT - Best Known (53−32, 53, s)-Nets in Base 64

Best Known (53−32, 53, s)-Nets in Base 64

(53−32, 53, 193)-Net over F₆₄ — Constructive and digital

Digital (21, 53, 193)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 16, 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, 37, 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]

(53−32, 53, 288)-Net in Base 64 — Constructive

(21, 53, 288)-net in base 64, using

31 times m-reduction [i] based on (21, 84, 288)-net in base 64, using
- base change [i] based on digital (9, 72, 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]

(53−32, 53, 342)-Net over F₆₄ — Digital

Digital (21, 53, 342)-net over F₆₄, using

t-expansion [i] based on digital (20, 53, 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]

(53−32, 53, 103785)-Net in Base 64 — Upper bound on s

There is no (21, 53, 103786)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 534033 748327 664173 446558 458224 008574 585302 918618 048298 642029 593532 769076 862964 712017 699891 183510 > 64⁵³ [i]