MinT - Best Known (252−43, 252, s)-Nets in Base 4

Best Known (252−43, 252, s)-Nets in Base 4

(252−43, 252, 1572)-Net over F₄ — Constructive and digital

Digital (209, 252, 1572)-net over F₄, using

1 times m-reduction [i] based on digital (209, 253, 1572)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 37, 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 (172, 216, 1539)-net over F₄, using
    - trace code for nets [i] based on digital (28, 72, 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]

(252−43, 252, 22565)-Net over F₄ — Digital

Digital (209, 252, 22565)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(252−43, 252, large)-Net in Base 4 — Upper bound on s

There is no (209, 252, large)-net in base 4, because

41 times m-reduction [i] would yield (209, 211, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]