MinT - Best Known (86−63, 86, s)-Nets in Base 16

Best Known (86−63, 86, s)-Nets in Base 16

(86−63, 86, 65)-Net over F₁₆ — Constructive and digital

Digital (23, 86, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 86, 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]

(86−63, 86, 76)-Net in Base 16 — Constructive

(23, 86, 76)-net in base 16, using

4 times m-reduction [i] based on (23, 90, 76)-net in base 16, using
- base change [i] based on digital (5, 72, 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]

(86−63, 86, 129)-Net over F₁₆ — Digital

Digital (23, 86, 129)-net over F₁₆, using

t-expansion [i] based on digital (19, 86, 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]

(86−63, 86, 1640)-Net in Base 16 — Upper bound on s

There is no (23, 86, 1641)-net in base 16, because

1 times m-reduction [i] would yield (23, 85, 1641)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 244461 358358 374869 867614 005499 945300 617912 840991 446189 845068 992074 526517 910890 217133 296400 031017 190816 > 16⁸⁵ [i]