MinT - Best Known (51−21, 51, s)-Nets in Base 4

Best Known (51−21, 51, s)-Nets in Base 4

(51−21, 51, 90)-Net over F₄ — Constructive and digital

Digital (30, 51, 90)-net over F₄, using

1 times m-reduction [i] based on digital (30, 52, 90)-net over F₄, using
- trace code for nets [i] based on digital (4, 26, 45)-net over F₁₆, using
  - net from sequence [i] based on digital (4, 44)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 4 and N(F) ≥ 45, using
      - a function field by GarcÃa and Quoos [i]

(51−21, 51, 91)-Net over F₄ — Digital

Digital (30, 51, 91)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(51−21, 51, 1537)-Net in Base 4 — Upper bound on s

There is no (30, 51, 1538)-net in base 4, because

1 times m-reduction [i] would yield (30, 50, 1538)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 267746 590270 935109 034151 330048 > 4⁵⁰ [i]