MinT - Best Known (65−57, 65, s)-Nets in Base 16

Best Known (65−57, 65, s)-Nets in Base 16

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

Digital (8, 65, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 65, 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]

(65−57, 65, 408)-Net in Base 16 — Upper bound on s

There is no (8, 65, 409)-net in base 16, because

5 times m-reduction [i] would yield (8, 60, 409)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 1 822339 856650 198012 911195 259157 124016 349729 875467 036119 357675 385495 075136 > 16⁶⁰ [i]