MinT - Best Known (57−31, 57, s)-Nets in Base 4

Best Known (57−31, 57, s)-Nets in Base 4

(57−31, 57, 36)-Net over F₄ — Constructive and digital

Digital (26, 57, 36)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (4, 19, 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, 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]

(57−31, 57, 55)-Net over F₄ — Digital

Digital (26, 57, 55)-net over F₄, using

net from sequence [i] based on digital (26, 54)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 26 and N(F) ≥ 55, using
  - Drinfeld modules of rank 1 [i]

(57−31, 57, 366)-Net in Base 4 — Upper bound on s

There is no (26, 57, 367)-net in base 4, because

1 times m-reduction [i] would yield (26, 56, 367)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 5216 815287 489952 795376 217779 484652 > 4⁵⁶ [i]