MinT - Best Known (13−9, 13, s)-Nets in Base 32

Best Known (13−9, 13, s)-Nets in Base 32

(13−9, 13, 66)-Net over F₃₂ — Constructive and digital

Digital (4, 13, 66)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (0, 4, 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, 9, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)

(13−9, 13, 71)-Net over F₃₂ — Digital

Digital (4, 13, 71)-net over F₃₂, using

net from sequence [i] based on digital (4, 70)-sequence over F₃₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 4 and N(F) ≥ 71, using
  - a curve from the manYPoints database [i]

(13−9, 13, 129)-Net in Base 32 — Constructive

(4, 13, 129)-net in base 32, using

1 times m-reduction [i] based on (4, 14, 129)-net in base 32, using
- base change [i] based on digital (0, 10, 129)-net over F₁₂₈, using
  - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
      - the rational function field F₁₂₈(x) [i]
    - Niederreiter sequence [i]

(13−9, 13, 2337)-Net in Base 32 — Upper bound on s

There is no (4, 13, 2338)-net in base 32, because

1 times m-reduction [i] would yield (4, 12, 2338)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 1 152983 996457 464984 > 32¹² [i]