MinT - Best Known (75−57, 75, s)-Nets in Base 32

Best Known (75−57, 75, s)-Nets in Base 32

(75−57, 75, 120)-Net over F₃₂ — Constructive and digital

Digital (18, 75, 120)-net over F₃₂, using

t-expansion [i] based on digital (11, 75, 120)-net over F₃₂, using
- net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
    - a function field by SÃ©mirat [i]

(75−57, 75, 128)-Net in Base 32 — Constructive

(18, 75, 128)-net in base 32, using

3 times m-reduction [i] based on (18, 78, 128)-net in base 32, using
- base change [i] based on digital (5, 65, 128)-net over F₆₄, using
  - net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
      - a function field by SÃ©mirat [i]

(75−57, 75, 161)-Net over F₃₂ — Digital

Digital (18, 75, 161)-net over F₃₂, using

net from sequence [i] based on digital (18, 160)-sequence over F₃₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 18 and N(F) ≥ 161, using
  - a curve from the manYPoints database [i]

(75−57, 75, 3449)-Net in Base 32 — Upper bound on s

There is no (18, 75, 3450)-net in base 32, because

1 times m-reduction [i] would yield (18, 74, 3450)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2418 449324 387439 076328 756404 364324 156814 668228 963974 140839 830085 466906 711683 670612 455560 064737 251073 839172 399464 > 32⁷⁴ [i]