MinT - Best Known (37, 37+51, s)-Nets in Base 64

Best Known (37, 37+51, s)-Nets in Base 64

(37, 37+51, 513)-Net over F₆₄ — Constructive and digital

Digital (37, 88, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 88, 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]

(37, 37+51, 540)-Net over F₆₄ — Digital

Digital (37, 88, 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]

(37, 37+51, 311726)-Net in Base 64 — Upper bound on s

There is no (37, 88, 311727)-net in base 64, because

1 times m-reduction [i] would yield (37, 87, 311727)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 13 729677 094038 429230 000085 704701 661984 985154 841693 785282 471909 153070 111854 758061 290401 565936 159369 452507 457705 091418 042297 868900 301352 427083 808554 165098 317916 > 64⁸⁷ [i]