MinT - Best Known (45−31, 45, s)-Nets in Base 64

Best Known (45−31, 45, s)-Nets in Base 64

(45−31, 45, 177)-Net over F₆₄ — Constructive and digital

Digital (14, 45, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 45, 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]

(45−31, 45, 257)-Net over F₆₄ — Digital

Digital (14, 45, 257)-net over F₆₄, using

t-expansion [i] based on digital (12, 45, 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]

(45−31, 45, 259)-Net in Base 64 — Constructive

(14, 45, 259)-net in base 64, using

3 times m-reduction [i] based on (14, 48, 259)-net in base 64, using
- base change [i] based on digital (2, 36, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(45−31, 45, 321)-Net in Base 64

(14, 45, 321)-net in base 64, using

3 times m-reduction [i] based on (14, 48, 321)-net in base 64, using
- base change [i] based on digital (2, 36, 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]

(45−31, 45, 20248)-Net in Base 64 — Upper bound on s

There is no (14, 45, 20249)-net in base 64, because

1 times m-reduction [i] would yield (14, 44, 20249)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 29 647655 302323 047189 820375 540946 420842 277569 684120 316913 742408 518704 816791 738256 > 64⁴⁴ [i]