MinT - Best Known (31, 31+99, s)-Nets in Base 16

Best Known (31, 31+99, s)-Nets in Base 16

(31, 31+99, 65)-Net over F₁₆ — Constructive and digital

Digital (31, 130, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 130, 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]

(31, 31+99, 76)-Net in Base 16 — Constructive

(31, 130, 76)-net in base 16, using

base change [i] based on digital (5, 104, 76)-net over F₃₂, using
- net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
    - a function field by SÃ©mirat [i]

(31, 31+99, 168)-Net over F₁₆ — Digital

Digital (31, 130, 168)-net over F₁₆, using

net from sequence [i] based on digital (31, 167)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 31 and N(F) ≥ 168, using
  - a curve from the manYPoints database [i]

(31, 31+99, 1857)-Net in Base 16 — Upper bound on s

There is no (31, 130, 1858)-net in base 16, because

1 times m-reduction [i] would yield (31, 129, 1858)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 217039 416607 195394 022994 168516 162895 078849 808463 125820 686433 829769 513180 013257 234857 721872 808048 011312 210234 416540 660046 831371 900979 047963 571162 228891 474356 > 16¹²⁹ [i]