MinT - Best Known (51−19, 51, s)-Nets in Base 4

Best Known (51−19, 51, s)-Nets in Base 4

(51−19, 51, 130)-Net over F₄ — Constructive and digital

Digital (32, 51, 130)-net over F₄, using

1 times m-reduction [i] based on digital (32, 52, 130)-net over F₄, using
- trace code for nets [i] based on digital (6, 26, 65)-net over F₁₆, using
  - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
      - the Hermitian function field over F₁₆ [i]

(51−19, 51, 131)-Net over F₄ — Digital

Digital (32, 51, 131)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(51−19, 51, 3050)-Net in Base 4 — Upper bound on s

There is no (32, 51, 3051)-net in base 4, because

1 times m-reduction [i] would yield (32, 50, 3051)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 268403 748644 841472 172073 814074 > 4⁵⁰ [i]