MinT - Best Known (18, 109, s)-Nets in Base 16

Best Known (18, 109, s)-Nets in Base 16

(18, 109, 65)-Net over F₁₆ — Constructive and digital

Digital (18, 109, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 109, 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, 109, 113)-Net over F₁₆ — Digital

Digital (18, 109, 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, 109, 887)-Net in Base 16 — Upper bound on s

There is no (18, 109, 888)-net in base 16, because

1 times m-reduction [i] would yield (18, 108, 888)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 11584 564250 423952 441301 038349 602340 735001 182304 617809 487452 146950 117690 820272 772739 573821 440490 868833 072411 927781 236943 667953 516526 > 16¹⁰⁸ [i]