MinT - Best Known (80−49, 80, s)-Nets in Base 32

Best Known (80−49, 80, s)-Nets in Base 32

(80−49, 80, 131)-Net over F₃₂ — Constructive and digital

Digital (31, 80, 131)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (0, 24, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 0 and N(F) ≥ 33, using
      - the rational function field F₃₂(x) [i]
    - Niederreiter sequence [i]
2. digital (7, 56, 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]

(80−49, 80, 258)-Net in Base 32 — Constructive

(31, 80, 258)-net in base 32, using

base change [i] based on digital (1, 50, 258)-net over F₂₅₆, using
- net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(80−49, 80, 273)-Net over F₃₂ — Digital

Digital (31, 80, 273)-net over F₃₂, using

t-expansion [i] based on digital (30, 80, 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]

(80−49, 80, 289)-Net in Base 32

(31, 80, 289)-net in base 32, using

base change [i] based on digital (1, 50, 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]

(80−49, 80, 28462)-Net in Base 32 — Upper bound on s

There is no (31, 80, 28463)-net in base 32, because

1 times m-reduction [i] would yield (31, 79, 28463)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 80761 524415 878849 889822 512477 270494 708277 414501 784763 708017 525337 028373 530312 912221 123868 512675 952547 971412 764814 891886 > 32⁷⁹ [i]