MinT - Best Known (31, 31+69, s)-Nets in Base 16

Best Known (31, 31+69, s)-Nets in Base 16

(31, 31+69, 65)-Net over F₁₆ — Constructive and digital

Digital (31, 100, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 100, 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+69, 120)-Net in Base 16 — Constructive

(31, 100, 120)-net in base 16, using

base change [i] based on digital (11, 80, 120)-net over F₃₂, using
- net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
    - a function field by SÃ©mirat [i]

(31, 31+69, 168)-Net over F₁₆ — Digital

Digital (31, 100, 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+69, 2875)-Net in Base 16 — Upper bound on s

There is no (31, 100, 2876)-net in base 16, because

1 times m-reduction [i] would yield (31, 99, 2876)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 162796 303173 349044 401690 022100 674355 867852 495832 161833 568032 002219 584820 527688 770973 262165 850376 958404 601134 659788 670861 > 16⁹⁹ [i]