MinT - Best Known (57−31, 57, s)-Nets in Base 32

Best Known (57−31, 57, s)-Nets in Base 32

(57−31, 57, 162)-Net over F₃₂ — Constructive and digital

Digital (26, 57, 162)-net over F₃₂, using

1 times m-reduction [i] based on digital (26, 58, 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, 39, 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]

(57−31, 57, 288)-Net in Base 32 — Constructive

(26, 57, 288)-net in base 32, using

32¹ times duplication [i] based on (25, 56, 288)-net in base 32, using
- base change [i] based on digital (9, 40, 288)-net over F₁₂₈, using
  - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
      - a function field by SÃ©mirat [i]

(57−31, 57, 292)-Net over F₃₂ — Digital

Digital (26, 57, 292)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(57−31, 57, 321)-Net in Base 32

(26, 57, 321)-net in base 32, using

7 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]

(57−31, 57, 86217)-Net in Base 32 — Upper bound on s

There is no (26, 57, 86218)-net in base 32, because

1 times m-reduction [i] would yield (26, 56, 86218)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 1 942944 466682 929970 393509 861058 418542 873575 918530 265537 839776 782085 206585 895116 012448 > 32⁵⁶ [i]