MinT - Best Known (251−43, 251, s)-Nets in Base 4

Best Known (251−43, 251, s)-Nets in Base 4

(251−43, 251, 1572)-Net over F₄ — Constructive and digital

Digital (208, 251, 1572)-net over F₄, using

4² times duplication [i] based on digital (206, 249, 1572)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 36, 33)-net over F₄, using
    - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
        an explicitly constructive algebraic function field [i]
  2. digital (170, 213, 1539)-net over F₄, using
    - trace code for nets [i] based on digital (28, 71, 513)-net over F₆₄, using
      - net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
        the Hermitian function field over F₆₄ [i]

(251−43, 251, 21833)-Net over F₄ — Digital

Digital (208, 251, 21833)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(251−43, 251, large)-Net in Base 4 — Upper bound on s

There is no (208, 251, large)-net in base 4, because

41 times m-reduction [i] would yield (208, 210, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]