MinT - Best Known (50−36, 50, s)-Nets in Base 64

Best Known (50−36, 50, s)-Nets in Base 64

(50−36, 50, 177)-Net over F₆₄ — Constructive and digital

Digital (14, 50, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 50, 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]

(50−36, 50, 257)-Net over F₆₄ — Digital

Digital (14, 50, 257)-net over F₆₄, using

t-expansion [i] based on digital (12, 50, 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]

(50−36, 50, 258)-Net in Base 64 — Constructive

(14, 50, 258)-net in base 64, using

2 times m-reduction [i] based on (14, 52, 258)-net in base 64, using
- base change [i] based on digital (1, 39, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(50−36, 50, 289)-Net in Base 64

(14, 50, 289)-net in base 64, using

2 times m-reduction [i] based on (14, 52, 289)-net in base 64, using
- base change [i] based on digital (1, 39, 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]

(50−36, 50, 12463)-Net in Base 64 — Upper bound on s

There is no (14, 50, 12464)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 2 037653 664135 833088 024026 665084 747640 876003 858841 990168 430449 218188 554097 600858 640690 379010 > 64⁵⁰ [i]