MinT - Best Known (81−63, 81, s)-Nets in Base 16

Best Known (81−63, 81, s)-Nets in Base 16

(81−63, 81, 65)-Net over F₁₆ — Constructive and digital

Digital (18, 81, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 81, 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]

(81−63, 81, 113)-Net over F₁₆ — Digital

Digital (18, 81, 113)-net over F₁₆, using

net from sequence [i] based on digital (18, 112)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 18 and N(F) ≥ 113, using
  - a curve from the manYPoints database [i]

(81−63, 81, 1043)-Net in Base 16 — Upper bound on s

There is no (18, 81, 1044)-net in base 16, because

1 times m-reduction [i] would yield (18, 80, 1044)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 192604 405327 999221 691743 243673 528494 872523 932252 351111 654566 310164 156917 390534 886606 788940 142336 > 16⁸⁰ [i]