MinT - Best Known (27−17, 27, s)-Nets in Base 16

Best Known (27−17, 27, s)-Nets in Base 16

(27−17, 27, 65)-Net over F₁₆ — Constructive and digital

Digital (10, 27, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 27, 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]

(27−17, 27, 80)-Net in Base 16 — Constructive

(10, 27, 80)-net in base 16, using

base change [i] based on digital (1, 18, 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]

(27−17, 27, 81)-Net over F₁₆ — Digital

Digital (10, 27, 81)-net over F₁₆, using

net from sequence [i] based on digital (10, 80)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 10 and N(F) ≥ 81, using
  - a curve from the manYPoints database [i]

(27−17, 27, 2051)-Net in Base 16 — Upper bound on s

There is no (10, 27, 2052)-net in base 16, because

1 times m-reduction [i] would yield (10, 26, 2052)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 20 297706 467217 244199 360926 518016 > 16²⁶ [i]