MinT - Best Known (12, 12+31, s)-Nets in Base 16

Best Known (12, 12+31, s)-Nets in Base 16

(12, 12+31, 65)-Net over F₁₆ — Constructive and digital

Digital (12, 43, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 43, 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]

(12, 12+31, 88)-Net over F₁₆ — Digital

Digital (12, 43, 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]

(12, 12+31, 999)-Net in Base 16 — Upper bound on s

There is no (12, 43, 1000)-net in base 16, because

1 times m-reduction [i] would yield (12, 42, 1000)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 377 285203 703618 385487 222937 297393 362378 295663 521876 > 16⁴² [i]