MinT - Best Known (172−44, 172, s)-Nets in Base 8

Best Known (172−44, 172, s)-Nets in Base 8

(172−44, 172, 1054)-Net over F₈ — Constructive and digital

Digital (128, 172, 1054)-net over F₈, using

1 times m-reduction [i] based on digital (128, 173, 1054)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 27, 28)-net over F₈, using
    - net from sequence [i] based on digital (5, 27)-sequence over F₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 5 and N(F) ≥ 28, using
        a function field by SÃ©mirat [i]
  2. digital (101, 146, 1026)-net over F₈, using
    - trace code for nets [i] based on digital (28, 73, 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]

(172−44, 172, 9901)-Net over F₈ — Digital

Digital (128, 172, 9901)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(172−44, 172, large)-Net in Base 8 — Upper bound on s

There is no (128, 172, large)-net in base 8, because

42 times m-reduction [i] would yield (128, 130, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]