MinT - Best Known (51−27, 51, s)-Nets in Base 4

Best Known (51−27, 51, s)-Nets in Base 4

(51−27, 51, 36)-Net over F₄ — Constructive and digital

Digital (24, 51, 36)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (4, 17, 15)-net over F₄, using
  - net from sequence [i] based on digital (4, 14)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 4 and N(F) ≥ 15, using
      - an explicitly constructive algebraic function field [i]
2. digital (7, 34, 21)-net over F₄, using
  - net from sequence [i] based on digital (7, 20)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 7 and N(F) ≥ 21, using
      - an explicitly constructive algebraic function field [i]

(51−27, 51, 49)-Net over F₄ — Digital

Digital (24, 51, 49)-net over F₄, using

net from sequence [i] based on digital (24, 48)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 24 and N(F) ≥ 49, using
  - a curve from the manYPoints database [i]

(51−27, 51, 380)-Net in Base 4 — Upper bound on s

There is no (24, 51, 381)-net in base 4, because

1 times m-reduction [i] would yield (24, 50, 381)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 287916 603907 415688 686082 468060 > 4⁵⁰ [i]