MinT - Best Known (41−19, 41, s)-Nets in Base 4

Best Known (41−19, 41, s)-Nets in Base 4

(41−19, 41, 48)-Net over F₄ — Constructive and digital

Digital (22, 41, 48)-net over F₄, using

1 times m-reduction [i] based on digital (22, 42, 48)-net over F₄, using
- trace code for nets [i] based on digital (1, 21, 24)-net over F₁₆, using
  - net from sequence [i] based on digital (1, 23)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 1 and N(F) ≥ 24, using
      - a function field by SÃ©mirat [i]

(41−19, 41, 54)-Net over F₄ — Digital

Digital (22, 41, 54)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(41−19, 41, 648)-Net in Base 4 — Upper bound on s

There is no (22, 41, 649)-net in base 4, because

1 times m-reduction [i] would yield (22, 40, 649)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 219798 390715 092548 480484 > 4⁴⁰ [i]