MinT - Best Known (31−13, 31, s)-Nets in Base 4

Best Known (31−13, 31, s)-Nets in Base 4

(31−13, 31, 66)-Net over F₄ — Constructive and digital

Digital (18, 31, 66)-net over F₄, using

1 times m-reduction [i] based on digital (18, 32, 66)-net over F₄, using
- trace code for nets [i] based on digital (2, 16, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
      - a function field by SÃ©mirat [i]

(31−13, 31, 1017)-Net in Base 4 — Upper bound on s

There is no (18, 31, 1018)-net in base 4, because

1 times m-reduction [i] would yield (18, 30, 1018)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 157056 902235 640620 > 4³⁰ [i]