MinT - Best Known (81−61, 81, s)-Nets in Base 16

Best Known (81−61, 81, s)-Nets in Base 16

(81−61, 81, 65)-Net over F₁₆ — Constructive and digital

Digital (20, 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]

(81−61, 81, 129)-Net over F₁₆ — Digital

Digital (20, 81, 129)-net over F₁₆, using

t-expansion [i] based on digital (19, 81, 129)-net over F₁₆, using
- net from sequence [i] based on digital (19, 128)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 19 and N(F) ≥ 129, using
    - a curve from the manYPoints database [i]

(81−61, 81, 1288)-Net in Base 16 — Upper bound on s

There is no (20, 81, 1289)-net in base 16, because

1 times m-reduction [i] would yield (20, 80, 1289)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 147153 147676 939105 854576 035642 019124 526529 671475 685044 098317 755223 314987 322359 823553 235785 538176 > 16⁸⁰ [i]