MinT - Best Known (51−45, 51, s)-Nets in Base 64

Best Known (51−45, 51, s)-Nets in Base 64

(51−45, 51, 128)-Net over F₆₄ — Constructive and digital

Digital (6, 51, 128)-net over F₆₄, using

t-expansion [i] based on digital (5, 51, 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]

(51−45, 51, 161)-Net over F₆₄ — Digital

Digital (6, 51, 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]

(51−45, 51, 1819)-Net in Base 64 — Upper bound on s

There is no (6, 51, 1820)-net in base 64, because

1 times m-reduction [i] would yield (6, 50, 1820)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 2 055957 950271 168271 876427 286474 195668 603328 740520 661727 582344 314499 537724 448940 268548 219834 > 64⁵⁰ [i]