MinT - Best Known (31−15, 31, s)-Nets in Base 4

Best Known (31−15, 31, s)-Nets in Base 4

(31−15, 31, 34)-Net over F₄ — Constructive and digital

Digital (16, 31, 34)-net over F₄, using

1 times m-reduction [i] based on digital (16, 32, 34)-net over F₄, using
- trace code for nets [i] based on digital (0, 16, 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]

(31−15, 31, 40)-Net over F₄ — Digital

Digital (16, 31, 40)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(31−15, 31, 423)-Net in Base 4 — Upper bound on s

There is no (16, 31, 424)-net in base 4, because

1 times m-reduction [i] would yield (16, 30, 424)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 166257 788257 496055 > 4³⁰ [i]