MinT - Best Known (18−9, 18, s)-Nets in Base 4

Best Known (18−9, 18, s)-Nets in Base 4

(18−9, 18, 34)-Net over F₄ — Constructive and digital

Digital (9, 18, 34)-net over F₄, using

trace code for nets [i] based on digital (0, 9, 17)-net over F₁₆, using
- net from sequence [i] based on digital (0, 16)-sequence over F₁₆, using
  - generalized Faure sequence [i]
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 0 and N(F) ≥ 17, using
    - the rational function field F₁₆(x) [i]
  - Niederreiter sequence [i]

(18−9, 18, 264)-Net in Base 4 — Upper bound on s

There is no (9, 18, 265)-net in base 4, because

1 times m-reduction [i] would yield (9, 17, 265)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 17364 122086 > 4¹⁷ [i]