MinT - Best Known (78−67, 78, s)-Nets in Base 16

Best Known (78−67, 78, s)-Nets in Base 16

(78−67, 78, 65)-Net over F₁₆ — Constructive and digital

Digital (11, 78, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 78, 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]

(78−67, 78, 81)-Net over F₁₆ — Digital

Digital (11, 78, 81)-net over F₁₆, using

t-expansion [i] based on digital (10, 78, 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]

(78−67, 78, 548)-Net in Base 16 — Upper bound on s

There is no (11, 78, 549)-net in base 16, because

1 times m-reduction [i] would yield (11, 77, 549)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 552 028206 976873 146301 999790 902672 991177 288134 790175 993313 746692 201702 434905 468371 470849 651756 > 16⁷⁷ [i]