MinT - Best Known (86−49, 86, s)-Nets in Base 64

Best Known (86−49, 86, s)-Nets in Base 64

(86−49, 86, 513)-Net over F₆₄ — Constructive and digital

Digital (37, 86, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 86, 513)-net over F₆₄, using
- net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
    - the Hermitian function field over F₆₄ [i]

(86−49, 86, 540)-Net over F₆₄ — Digital

Digital (37, 86, 540)-net over F₆₄, using

net from sequence [i] based on digital (37, 539)-sequence over F₆₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 37 and N(F) ≥ 540, using
  - a curve from the manYPoints database [i]

(86−49, 86, 388057)-Net in Base 64 — Upper bound on s

There is no (37, 86, 388058)-net in base 64, because

1 times m-reduction [i] would yield (37, 85, 388058)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 3352 015454 450895 189890 046439 150190 864185 753504 159312 874851 657921 519740 669203 468733 065581 761665 759331 418417 906150 768438 149396 738703 156968 758340 726974 822149 > 64⁸⁵ [i]