MinT - Best Known (9−7, 9, s)-Nets in Base 32

Best Known (9−7, 9, s)-Nets in Base 32

(9−7, 9, 44)-Net over F₃₂ — Constructive and digital

Digital (2, 9, 44)-net over F₃₂, using

t-expansion [i] based on digital (1, 9, 44)-net over F₃₂, using
- net from sequence [i] based on digital (1, 43)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 1 and N(F) ≥ 44, using
    - a function field by SÃ©mirat [i]

(9−7, 9, 53)-Net over F₃₂ — Digital

Digital (2, 9, 53)-net over F₃₂, using

net from sequence [i] based on digital (2, 52)-sequence over F₃₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 2 and N(F) ≥ 53, using
  - sharp bound on the number of rational points on curves with genus 2 [i]

(9−7, 9, 65)-Net in Base 32 — Constructive

(2, 9, 65)-net in base 32, using

3 times m-reduction [i] based on (2, 12, 65)-net in base 32, using
- base change [i] based on digital (0, 10, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
      - the rational function field F₆₄(x) [i]
    - Niederreiter sequence [i]

(9−7, 9, 603)-Net in Base 32 — Upper bound on s

There is no (2, 9, 604)-net in base 32, because

1 times m-reduction [i] would yield (2, 8, 604)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 1 100035 758323 > 32⁸ [i]