MinT - Best Known (81−59, 81, s)-Nets in Base 16

Best Known (81−59, 81, s)-Nets in Base 16

(81−59, 81, 65)-Net over F₁₆ — Constructive and digital

Digital (22, 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]

(81−59, 81, 76)-Net in Base 16 — Constructive

(22, 81, 76)-net in base 16, using

4 times m-reduction [i] based on (22, 85, 76)-net in base 16, using
- base change [i] based on digital (5, 68, 76)-net over F₃₂, using
  - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
      - a function field by SÃ©mirat [i]

(81−59, 81, 129)-Net over F₁₆ — Digital

Digital (22, 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]

(81−59, 81, 1616)-Net in Base 16 — Upper bound on s

There is no (22, 81, 1617)-net in base 16, because

1 times m-reduction [i] would yield (22, 80, 1617)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 167165 223713 923288 664452 284290 648645 796801 964644 232180 751354 866690 869234 745166 619120 207174 917696 > 16⁸⁰ [i]