MinT - Best Known (178−39, 178, s)-Nets in Base 2

Best Known (178−39, 178, s)-Nets in Base 2

(178−39, 178, 198)-Net over F₂ — Constructive and digital

Digital (139, 178, 198)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (0, 19, 3)-net over F₂, using
  - net from sequence [i] based on digital (0, 2)-sequence over F₂, using
    - Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 0 and N(F) ≥ 3, using
      - the rational function field F₂(x) [i]
    - Niederreiter sequence [i]
    - Sobol sequence [i]
2. digital (120, 159, 195)-net over F₂, using
  - trace code for nets [i] based on digital (14, 53, 65)-net over F₈, using
    - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
        the Suzuki function field over F₈ [i]

(178−39, 178, 361)-Net over F₂ — Digital

Digital (139, 178, 361)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(178−39, 178, 5025)-Net in Base 2 — Upper bound on s

There is no (139, 178, 5026)-net in base 2, because

1 times m-reduction [i] would yield (139, 177, 5026)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 192247 159581 620239 098646 671219 650726 382582 306024 339368 > 2¹⁷⁷ [i]