MinT - Best Known (67−57, 67, s)-Nets in Base 16

Best Known (67−57, 67, s)-Nets in Base 16

(67−57, 67, 65)-Net over F₁₆ — Constructive and digital

Digital (10, 67, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 67, 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]

(67−57, 67, 81)-Net over F₁₆ — Digital

Digital (10, 67, 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]

(67−57, 67, 503)-Net in Base 16 — Upper bound on s

There is no (10, 67, 504)-net in base 16, because

1 times m-reduction [i] would yield (10, 66, 504)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 29 993516 913949 962085 490977 761174 537003 300360 344455 147758 975559 707956 885764 101556 > 16⁶⁶ [i]