MinT - Best Known (72−55, 72, s)-Nets in Base 16

Best Known (72−55, 72, s)-Nets in Base 16

(72−55, 72, 65)-Net over F₁₆ — Constructive and digital

Digital (17, 72, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 72, 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]

(72−55, 72, 112)-Net over F₁₆ — Digital

Digital (17, 72, 112)-net over F₁₆, using

net from sequence [i] based on digital (17, 111)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 17 and N(F) ≥ 112, using
  - a curve from the manYPoints database [i]

(72−55, 72, 1053)-Net in Base 16 — Upper bound on s

There is no (17, 72, 1054)-net in base 16, because

1 times m-reduction [i] would yield (17, 71, 1054)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 31 413753 904927 829886 963065 373210 668265 810947 572763 288754 741852 335849 884728 722476 904746 > 16⁷¹ [i]