MinT - Best Known (31, 31+45, s)-Nets in Base 32

Best Known (31, 31+45, s)-Nets in Base 32

(31, 31+45, 142)-Net over F₃₂ — Constructive and digital

Digital (31, 76, 142)-net over F₃₂, using

1 times m-reduction [i] based on digital (31, 77, 142)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (1, 24, 44)-net over F₃₂, using
    - net from sequence [i] based on digital (1, 43)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 1 and N(F) ≥ 44, using
        a function field by SÃ©mirat [i]
  2. digital (7, 53, 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]

(31, 31+45, 273)-Net over F₃₂ — Digital

Digital (31, 76, 273)-net over F₃₂, using

t-expansion [i] based on digital (30, 76, 273)-net over F₃₂, using
- net from sequence [i] based on digital (30, 272)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 30 and N(F) ≥ 273, using
    - a curve from the manYPoints database [i]

(31, 31+45, 288)-Net in Base 32 — Constructive

(31, 76, 288)-net in base 32, using

1 times m-reduction [i] based on (31, 77, 288)-net in base 32, using
- base change [i] based on digital (9, 55, 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]

(31, 31+45, 321)-Net in Base 32

(31, 76, 321)-net in base 32, using

1 times m-reduction [i] based on (31, 77, 321)-net in base 32, using
- base change [i] based on (9, 55, 321)-net in base 128, using
  - 1 times m-reduction [i] based on (9, 56, 321)-net in base 128, using
    - base change [i] based on digital (2, 49, 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]

(31, 31+45, 39496)-Net in Base 32 — Upper bound on s

There is no (31, 76, 39497)-net in base 32, because

1 times m-reduction [i] would yield (31, 75, 39497)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 76968 354393 336638 697061 815351 719466 254229 190859 118117 591555 672759 880289 983613 101191 756860 587366 823793 235943 531088 > 32⁷⁵ [i]