MinT - Best Known (4, 21, s)-Nets in Base 32

Best Known (4, 21, s)-Nets in Base 32

(4, 21, 64)-Net over F₃₂ — Constructive and digital

Digital (4, 21, 64)-net over F₃₂, using

t-expansion [i] based on digital (3, 21, 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]

(4, 21, 65)-Net in Base 32 — Constructive

(4, 21, 65)-net in base 32, using

3 times m-reduction [i] based on (4, 24, 65)-net in base 32, using
- base change [i] based on digital (0, 20, 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]

(4, 21, 71)-Net over F₃₂ — Digital

Digital (4, 21, 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]

(4, 21, 699)-Net in Base 32 — Upper bound on s

There is no (4, 21, 700)-net in base 32, because

1 times m-reduction [i] would yield (4, 20, 700)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 1 272726 738643 392763 044662 399191 > 32²⁰ [i]