MinT - Best Known (31−17, 31, s)-Nets in Base 32

Best Known (31−17, 31, s)-Nets in Base 32

(31−17, 31, 128)-Net over F₃₂ — Constructive and digital

Digital (14, 31, 128)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (3, 11, 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 (3, 20, 64)-net over F₃₂, using
  - net from sequence [i] based on digital (3, 63)-sequence over F₃₂ (see above)

(31−17, 31, 189)-Net over F₃₂ — Digital

Digital (14, 31, 189)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(31−17, 31, 259)-Net in Base 32 — Constructive

(14, 31, 259)-net in base 32, using

1 times m-reduction [i] based on (14, 32, 259)-net in base 32, using
- base change [i] based on digital (2, 20, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(31−17, 31, 321)-Net in Base 32

(14, 31, 321)-net in base 32, using

1 times m-reduction [i] based on (14, 32, 321)-net in base 32, using
- base change [i] based on digital (2, 20, 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]

(31−17, 31, 53531)-Net in Base 32 — Upper bound on s

There is no (14, 31, 53532)-net in base 32, because

1 times m-reduction [i] would yield (14, 30, 53532)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 1427 330927 200106 942448 420961 530294 867846 247435 > 32³⁰ [i]