MinT - Best Known (18, 77, s)-Nets in Base 16

Best Known (18, 77, s)-Nets in Base 16

(18, 77, 65)-Net over F₁₆ — Constructive and digital

Digital (18, 77, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 77, 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]

(18, 77, 113)-Net over F₁₆ — Digital

Digital (18, 77, 113)-net over F₁₆, using

net from sequence [i] based on digital (18, 112)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 18 and N(F) ≥ 113, using
  - a curve from the manYPoints database [i]

(18, 77, 1097)-Net in Base 16 — Upper bound on s

There is no (18, 77, 1098)-net in base 16, because

1 times m-reduction [i] would yield (18, 76, 1098)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 32 989733 212753 171154 676197 974006 189403 096263 248593 237205 244019 858240 539945 312042 969477 275856 > 16⁷⁶ [i]