MinT - Best Known (21−15, 21, s)-Nets in Base 64

Best Known (21−15, 21, s)-Nets in Base 64

(21−15, 21, 128)-Net over F₆₄ — Constructive and digital

Digital (6, 21, 128)-net over F₆₄, using

t-expansion [i] based on digital (5, 21, 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]

(21−15, 21, 161)-Net over F₆₄ — Digital

Digital (6, 21, 161)-net over F₆₄, using

net from sequence [i] based on digital (6, 160)-sequence over F₆₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 6 and N(F) ≥ 161, using
  - a curve from the manYPoints database [i]

(21−15, 21, 257)-Net in Base 64 — Constructive

(6, 21, 257)-net in base 64, using

3 times m-reduction [i] based on (6, 24, 257)-net in base 64, using
- base change [i] based on digital (0, 18, 257)-net over F₂₅₆, using
  - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
      - the rational function field F₂₅₆(x) [i]
    - Niederreiter sequence [i]

(21−15, 21, 7761)-Net in Base 64 — Upper bound on s

There is no (6, 21, 7762)-net in base 64, because

1 times m-reduction [i] would yield (6, 20, 7762)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 1 330414 326389 604588 154578 943931 345320 > 64²⁰ [i]