MinT - Best Known (59−33, 59, s)-Nets in Base 16

Best Known (59−33, 59, s)-Nets in Base 16

(59−33, 59, 110)-Net over F₁₆ — Constructive and digital

Digital (26, 59, 110)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (4, 20, 45)-net over F₁₆, using
  - net from sequence [i] based on digital (4, 44)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 4 and N(F) ≥ 45, using
      - a function field by GarcÃa and Quoos [i]
2. digital (6, 39, 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]

(59−33, 59, 129)-Net in Base 16 — Constructive

(26, 59, 129)-net in base 16, using

1 times m-reduction [i] based on (26, 60, 129)-net in base 16, using
- base change [i] based on (6, 40, 129)-net in base 64, using
  - 2 times m-reduction [i] based on (6, 42, 129)-net in base 64, using
    - base change [i] based on digital (0, 36, 129)-net over F₁₂₈, using
      - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
        the rational function field F₁₂₈(x) [i]
        Niederreiter sequence [i]

(59−33, 59, 150)-Net over F₁₆ — Digital

Digital (26, 59, 150)-net over F₁₆, using

net from sequence [i] based on digital (26, 149)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 26 and N(F) ≥ 150, using
  - a curve from the manYPoints database [i]

(59−33, 59, 161)-Net in Base 16

(26, 59, 161)-net in base 16, using

1 times m-reduction [i] based on (26, 60, 161)-net in base 16, using
- base change [i] based on digital (6, 40, 161)-net over F₆₄, using
  - net from sequence [i] based on digital (6, 160)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 6 and N(F) ≥ 161, using
      - a curve from the manYPoints database [i]

(59−33, 59, 10496)-Net in Base 16 — Upper bound on s

There is no (26, 59, 10497)-net in base 16, because

1 times m-reduction [i] would yield (26, 58, 10497)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 6911 162018 797569 718086 332021 506367 790417 093367 323577 826468 128778 444656 > 16⁵⁸ [i]