MinT - Best Known (77−68, 77, s)-Nets in Base 16

Best Known (77−68, 77, s)-Nets in Base 16

(77−68, 77, 65)-Net over F₁₆ — Constructive and digital

Digital (9, 77, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 77, 65)-net over F₁₆, using
- net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
    - the Hermitian function field over F₁₆ [i]

(77−68, 77, 72)-Net over F₁₆ — Digital

Digital (9, 77, 72)-net over F₁₆, using

net from sequence [i] based on digital (9, 71)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 9 and N(F) ≥ 72, using
  - a curve from the manYPoints database [i]

(77−68, 77, 454)-Net in Base 16 — Upper bound on s

There is no (9, 77, 455)-net in base 16, because

12 times m-reduction [i] would yield (9, 65, 455)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 1 868080 606814 873652 666598 645088 383016 364705 288513 351712 006504 524993 386173 089976 > 16⁶⁵ [i]