MinT - Best Known (64−47, 64, s)-Nets in Base 64

Best Known (64−47, 64, s)-Nets in Base 64

(64−47, 64, 177)-Net over F₆₄ — Constructive and digital

Digital (17, 64, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 64, 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]

(64−47, 64, 258)-Net in Base 64 — Constructive

(17, 64, 258)-net in base 64, using

base change [i] based on digital (1, 48, 258)-net over F₂₅₆, using
- net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(64−47, 64, 267)-Net over F₆₄ — Digital

Digital (17, 64, 267)-net over F₆₄, using

t-expansion [i] based on digital (16, 64, 267)-net over F₆₄, using
- net from sequence [i] based on digital (16, 266)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 16 and N(F) ≥ 267, using
    - a curve from the manYPoints database [i]

(64−47, 64, 289)-Net in Base 64

(17, 64, 289)-net in base 64, using

base change [i] based on digital (1, 48, 289)-net over F₂₅₆, using
- net from sequence [i] based on digital (1, 288)-sequence over F₂₅₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 1 and N(F) ≥ 289, using
    - sharp bound on the number of rational points on curves with genus 1 [i]

(64−47, 64, 13246)-Net in Base 64 — Upper bound on s

There is no (17, 64, 13247)-net in base 64, because

1 times m-reduction [i] would yield (17, 63, 13247)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 616036 035410 105300 495558 071424 265685 230191 334915 121423 663242 089090 680143 779739 520799 403751 540321 961021 334613 701688 > 64⁶³ [i]