MinT - Best Known (45−32, 45, s)-Nets in Base 64

Best Known (45−32, 45, s)-Nets in Base 64

(45−32, 45, 177)-Net over F₆₄ — Constructive and digital

Digital (13, 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]

(45−32, 45, 257)-Net over F₆₄ — Digital

Digital (13, 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]

(45−32, 45, 258)-Net in Base 64 — Constructive

(13, 45, 258)-net in base 64, using

3 times m-reduction [i] based on (13, 48, 258)-net in base 64, using
- base change [i] based on digital (1, 36, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(45−32, 45, 289)-Net in Base 64

(13, 45, 289)-net in base 64, using

3 times m-reduction [i] based on (13, 48, 289)-net in base 64, using
- base change [i] based on digital (1, 36, 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]

(45−32, 45, 12966)-Net in Base 64 — Upper bound on s

There is no (13, 45, 12967)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 1898 518269 772751 387793 877068 549553 059590 176232 386841 294463 507929 849521 174391 223574 > 64⁴⁵ [i]