MinT - Best Known (56−31, 56, s)-Nets in Base 4

Best Known (56−31, 56, s)-Nets in Base 4

(56−31, 56, 35)-Net over F₄ — Constructive and digital

Digital (25, 56, 35)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 18, 14)-net over F₄, using
  - net from sequence [i] based on digital (3, 13)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 3 and N(F) ≥ 14, using
      - an explicitly constructive algebraic function field [i]
2. digital (7, 38, 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]

(56−31, 56, 51)-Net over F₄ — Digital

Digital (25, 56, 51)-net over F₄, using

net from sequence [i] based on digital (25, 50)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 25 and N(F) ≥ 51, using
  - Drinfeld modules of rank 1 [i]

(56−31, 56, 333)-Net in Base 4 — Upper bound on s

There is no (25, 56, 334)-net in base 4, because

1 times m-reduction [i] would yield (25, 55, 334)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1328 809784 818609 610657 829023 878288 > 4⁵⁵ [i]