MinT - Best Known (19, 19+41, s)-Nets in Base 16

Best Known (19, 19+41, s)-Nets in Base 16

(19, 19+41, 65)-Net over F₁₆ — Constructive and digital

Digital (19, 60, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 60, 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]

(19, 19+41, 98)-Net in Base 16 — Constructive

(19, 60, 98)-net in base 16, using

base change [i] based on digital (7, 48, 98)-net over F₃₂, using
- net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
    - a function field by SÃ©mirat [i]

(19, 19+41, 129)-Net over F₁₆ — Digital

Digital (19, 60, 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]

(19, 19+41, 1963)-Net in Base 16 — Upper bound on s

There is no (19, 60, 1964)-net in base 16, because

1 times m-reduction [i] would yield (19, 59, 1964)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 111235 274325 769080 155008 643365 856129 275360 309310 914407 062090 942937 906076 > 16⁵⁹ [i]