MinT - Best Known (71−43, 71, s)-Nets in Base 32

Best Known (71−43, 71, s)-Nets in Base 32

(71−43, 71, 131)-Net over F₃₂ — Constructive and digital

Digital (28, 71, 131)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (0, 21, 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, 50, 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]

(71−43, 71, 257)-Net over F₃₂ — Digital

Digital (28, 71, 257)-net over F₃₂, using

net from sequence [i] based on digital (28, 256)-sequence over F₃₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 28 and N(F) ≥ 257, using
  - a curve from the manYPoints database [i]

(71−43, 71, 258)-Net in Base 32 — Constructive

(28, 71, 258)-net in base 32, using

1 times m-reduction [i] based on (28, 72, 258)-net in base 32, using
- base change [i] based on digital (1, 45, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(71−43, 71, 289)-Net in Base 32

(28, 71, 289)-net in base 32, using

1 times m-reduction [i] based on (28, 72, 289)-net in base 32, using
- base change [i] based on digital (1, 45, 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]

(71−43, 71, 29116)-Net in Base 32 — Upper bound on s

There is no (28, 71, 29117)-net in base 32, because

1 times m-reduction [i] would yield (28, 70, 29117)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2294 427653 143959 166957 978314 678376 115814 392644 728470 309294 936772 319688 609229 476839 348778 599096 916790 140776 > 32⁷⁰ [i]