MinT - Best Known (70−51, 70, s)-Nets in Base 16

Best Known (70−51, 70, s)-Nets in Base 16

(70−51, 70, 65)-Net over F₁₆ — Constructive and digital

Digital (19, 70, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 70, 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]

(70−51, 70, 76)-Net in Base 16 — Constructive

(19, 70, 76)-net in base 16, using

base change [i] based on digital (5, 56, 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]

(70−51, 70, 129)-Net over F₁₆ — Digital

Digital (19, 70, 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]

(70−51, 70, 1414)-Net in Base 16 — Upper bound on s

There is no (19, 70, 1415)-net in base 16, because

1 times m-reduction [i] would yield (19, 69, 1415)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 121522 721824 911172 582649 469264 876844 829637 875439 698543 642004 342726 404185 434015 849376 > 16⁶⁹ [i]