MinT - Best Known (10−7, 10, s)-Nets in Base 32

Best Known (10−7, 10, s)-Nets in Base 32

(10−7, 10, 66)-Net over F₃₂ — Constructive and digital

Digital (3, 10, 66)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (0, 3, 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 (0, 7, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)

(10−7, 10, 80)-Net in Base 32 — Constructive

(3, 10, 80)-net in base 32, using

2 times m-reduction [i] based on (3, 12, 80)-net in base 32, using
- base change [i] based on digital (1, 10, 80)-net over F₆₄, using
  - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
      - a function field by SÃ©mirat [i]

(10−7, 10, 81)-Net in Base 32

(3, 10, 81)-net in base 32, using

2 times m-reduction [i] based on (3, 12, 81)-net in base 32, using
- base change [i] based on digital (1, 10, 81)-net over F₆₄, using
  - net from sequence [i] based on digital (1, 80)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 81, using
      - sharp bound on the number of rational points on curves with genus 1 [i]

(10−7, 10, 1919)-Net in Base 32 — Upper bound on s

There is no (3, 10, 1920)-net in base 32, because

1 times m-reduction [i] would yield (3, 9, 1920)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 35 203140 297281 > 32⁹ [i]