MinT - Best Known (212−36, 212, s)-Nets in Base 4

Best Known (212−36, 212, s)-Nets in Base 4

(212−36, 212, 1549)-Net over F₄ — Constructive and digital

Digital (176, 212, 1549)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (2, 20, 10)-net over F₄, using
  - net from sequence [i] based on digital (2, 9)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 2 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
2. digital (156, 192, 1539)-net over F₄, using
  - trace code for nets [i] based on digital (28, 64, 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]

(212−36, 212, 20572)-Net over F₄ — Digital

Digital (176, 212, 20572)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(212−36, 212, large)-Net in Base 4 — Upper bound on s

There is no (176, 212, large)-net in base 4, because

34 times m-reduction [i] would yield (176, 178, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]