MinT - Best Known (31−19, 31, s)-Nets in Base 64

Best Known (31−19, 31, s)-Nets in Base 64

(31−19, 31, 177)-Net over F₆₄ — Constructive and digital

Digital (12, 31, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 31, 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]

(31−19, 31, 257)-Net over F₆₄ — Digital

Digital (12, 31, 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]

(31−19, 31, 261)-Net in Base 64 — Constructive

(12, 31, 261)-net in base 64, using

1 times m-reduction [i] based on (12, 32, 261)-net in base 64, using
- base change [i] based on digital (4, 24, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(31−19, 31, 321)-Net in Base 64

(12, 31, 321)-net in base 64, using

9 times m-reduction [i] based on (12, 40, 321)-net in base 64, using
- base change [i] based on digital (2, 30, 321)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 320)-sequence over F₂₅₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 2 and N(F) ≥ 321, using
      - sharp bound on the number of rational points on curves with genus 2 [i]

(31−19, 31, 69021)-Net in Base 64 — Upper bound on s

There is no (12, 31, 69022)-net in base 64, because

1 times m-reduction [i] would yield (12, 30, 69022)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 1 532584 265030 829298 698779 208779 763812 914315 599169 261939 > 64³⁰ [i]