MinT - Best Known (31−19, 31, s)-Nets in Base 16

Best Known (31−19, 31, s)-Nets in Base 16

(31−19, 31, 65)-Net over F₁₆ — Constructive and digital

Digital (12, 31, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 31, 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]

(31−19, 31, 80)-Net in Base 16 — Constructive

(12, 31, 80)-net in base 16, using

2 times m-reduction [i] based on (12, 33, 80)-net in base 16, using
- base change [i] based on digital (1, 22, 80)-net over F₆₄, using
  - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
      - a function field by SÃ©mirat [i]

(31−19, 31, 88)-Net over F₁₆ — Digital

Digital (12, 31, 88)-net over F₁₆, using

net from sequence [i] based on digital (12, 87)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 12 and N(F) ≥ 88, using
  - a curve from the manYPoints database [i]

(31−19, 31, 2849)-Net in Base 16 — Upper bound on s

There is no (12, 31, 2850)-net in base 16, because

1 times m-reduction [i] would yield (12, 30, 2850)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 1 333407 292704 587813 369177 342827 267876 > 16³⁰ [i]