MinT - Best Known (90−77, 90, s)-Nets in Base 16

Best Known (90−77, 90, s)-Nets in Base 16

(90−77, 90, 65)-Net over F₁₆ — Constructive and digital

Digital (13, 90, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 90, 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]

(90−77, 90, 97)-Net over F₁₆ — Digital

Digital (13, 90, 97)-net over F₁₆, using

net from sequence [i] based on digital (13, 96)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 13 and N(F) ≥ 97, using
  - a curve from the manYPoints database [i]

(90−77, 90, 641)-Net in Base 16 — Upper bound on s

There is no (13, 90, 642)-net in base 16, because

1 times m-reduction [i] would yield (13, 89, 642)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 153883 940089 464001 581340 802873 805527 650045 429300 542942 666725 214088 013648 187811 094773 517258 930208 711813 025216 > 16⁸⁹ [i]