MinT - Best Known (227−72, 227, s)-Nets in Base 2

Best Known (227−72, 227, s)-Nets in Base 2

(227−72, 227, 112)-Net over F₂ — Constructive and digital

Digital (155, 227, 112)-net over F₂, using

17 times m-reduction [i] based on digital (155, 244, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 122, 56)-net over F₄, using
  - net from sequence [i] based on digital (33, 55)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
      - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(227−72, 227, 174)-Net over F₂ — Digital

Digital (155, 227, 174)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(227−72, 227, 1077)-Net in Base 2 — Upper bound on s

There is no (155, 227, 1078)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 221 514153 193723 457146 879271 172692 499473 926943 424495 803732 079019 910975 > 2²²⁷ [i]