MinT - Best Known (30, 30+47, s)-Nets in Base 32

Best Known (30, 30+47, s)-Nets in Base 32

(30, 30+47, 131)-Net over F₃₂ — Constructive and digital

Digital (30, 77, 131)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (0, 23, 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, 54, 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]

(30, 30+47, 258)-Net in Base 32 — Constructive

(30, 77, 258)-net in base 32, using

base change [i] based on (8, 55, 258)-net in base 128, using
- 1 times m-reduction [i] based on (8, 56, 258)-net in base 128, using
  - base change [i] based on digital (1, 49, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(30, 30+47, 273)-Net over F₃₂ — Digital

Digital (30, 77, 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]

(30, 30+47, 289)-Net in Base 32

(30, 77, 289)-net in base 32, using

base change [i] based on (8, 55, 289)-net in base 128, using
- 1 times m-reduction [i] based on (8, 56, 289)-net in base 128, using
  - base change [i] based on digital (1, 49, 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]

(30, 30+47, 28605)-Net in Base 32 — Upper bound on s

There is no (30, 77, 28606)-net in base 32, because

1 times m-reduction [i] would yield (30, 76, 28606)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2 462847 430465 597053 304430 547361 978146 343198 054992 697336 769218 675770 882325 328076 340814 130739 579202 912804 125268 587164 > 32⁷⁶ [i]