MinT - Best Known (59−33, 59, s)-Nets in Base 32

Best Known (59−33, 59, s)-Nets in Base 32

(59−33, 59, 162)-Net over F₃₂ — Constructive and digital

Digital (26, 59, 162)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (3, 19, 64)-net over F₃₂, using
  - net from sequence [i] based on digital (3, 63)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 3 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]
2. digital (7, 40, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
      - a function field by SÃ©mirat [i]

(59−33, 59, 250)-Net over F₃₂ — Digital

Digital (26, 59, 250)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(59−33, 59, 260)-Net in Base 32 — Constructive

(26, 59, 260)-net in base 32, using

1 times m-reduction [i] based on (26, 60, 260)-net in base 32, using
- base change [i] based on (16, 50, 260)-net in base 64, using
  - 2 times m-reduction [i] based on (16, 52, 260)-net in base 64, using
    - base change [i] based on digital (3, 39, 260)-net over F₂₅₆, using
      - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(59−33, 59, 321)-Net in Base 32

(26, 59, 321)-net in base 32, using

5 times m-reduction [i] based on (26, 64, 321)-net in base 32, using
- base change [i] based on digital (2, 40, 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]

(59−33, 59, 62703)-Net in Base 32 — Upper bound on s

There is no (26, 59, 62704)-net in base 32, because

1 times m-reduction [i] would yield (26, 58, 62704)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 1989 663770 052748 521381 425239 324250 999243 040995 123963 070760 093419 827636 699523 901384 110870 > 32⁵⁸ [i]