MinT - Best Known (60−33, 60, s)-Nets in Base 16

Best Known (60−33, 60, s)-Nets in Base 16

(60−33, 60, 114)-Net over F₁₆ — Constructive and digital

Digital (27, 60, 114)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (5, 21, 49)-net over F₁₆, using
  - net from sequence [i] based on digital (5, 48)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 5 and N(F) ≥ 49, using
      - a function field by SÃ©mirat [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]

(60−33, 60, 156)-Net over F₁₆ — Digital

Digital (27, 60, 156)-net over F₁₆, using

net from sequence [i] based on digital (27, 155)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 27 and N(F) ≥ 156, using
  - a curve from the manYPoints database [i]

(60−33, 60, 177)-Net in Base 16 — Constructive

(27, 60, 177)-net in base 16, using

base change [i] based on digital (7, 40, 177)-net over F₆₄, using
- net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
    - a function field by GarcÃa and Quoos [i]

(60−33, 60, 12483)-Net in Base 16 — Upper bound on s

There is no (27, 60, 12484)-net in base 16, because

1 times m-reduction [i] would yield (27, 59, 12484)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 110475 635895 916298 996210 315139 835155 556703 684600 836186 299622 292805 658036 > 16⁵⁹ [i]