MinT - Best Known (39−19, 39, s)-Nets in Base 4

Best Known (39−19, 39, s)-Nets in Base 4

(39−19, 39, 36)-Net over F₄ — Constructive and digital

Digital (20, 39, 36)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (4, 13, 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, 26, 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]

(39−19, 39, 46)-Net over F₄ — Digital

Digital (20, 39, 46)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(39−19, 39, 474)-Net in Base 4 — Upper bound on s

There is no (20, 39, 475)-net in base 4, because

1 times m-reduction [i] would yield (20, 38, 475)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 76107 673162 530442 268686 > 4³⁸ [i]