MinT - Best Known (27, 90, s)-Nets in Base 16

Best Known (27, 90, s)-Nets in Base 16

(27, 90, 65)-Net over F₁₆ — Constructive and digital

Digital (27, 90, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 90, 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]

(27, 90, 104)-Net in Base 16 — Constructive

(27, 90, 104)-net in base 16, using

base change [i] based on digital (9, 72, 104)-net over F₃₂, using
- net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
    - a function field by SÃ©mirat [i]

(27, 90, 156)-Net over F₁₆ — Digital

Digital (27, 90, 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]

(27, 90, 2353)-Net in Base 16 — Upper bound on s

There is no (27, 90, 2354)-net in base 16, because

1 times m-reduction [i] would yield (27, 89, 2354)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 146855 211944 589192 937714 071698 783616 398438 178440 425645 711550 739536 265742 530389 086362 944805 312662 274200 862736 > 16⁸⁹ [i]