MinT - Best Known (41−21, 41, s)-Nets in Base 4

Best Known (41−21, 41, s)-Nets in Base 4

(41−21, 41, 35)-Net over F₄ — Constructive and digital

Digital (20, 41, 35)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 13, 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, 28, 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]

(41−21, 41, 41)-Net over F₄ — Digital

Digital (20, 41, 41)-net over F₄, using

t-expansion [i] based on digital (18, 41, 41)-net over F₄, using
- net from sequence [i] based on digital (18, 40)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 18 and N(F) ≥ 41, using
    - Drinfeld modules of rank 1 [i]

(41−21, 41, 378)-Net in Base 4 — Upper bound on s

There is no (20, 41, 379)-net in base 4, because

1 times m-reduction [i] would yield (20, 40, 379)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 218974 555429 855949 294211 > 4⁴⁰ [i]