MinT - Best Known (227−80, 227, s)-Nets in Base 2

Best Known (227−80, 227, s)-Nets in Base 2

(227−80, 227, 112)-Net over F₂ — Constructive and digital

Digital (147, 227, 112)-net over F₂, using

1 times m-reduction [i] based on digital (147, 228, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 114, 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]

(227−80, 227, 137)-Net over F₂ — Digital

Digital (147, 227, 137)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(227−80, 227, 748)-Net in Base 2 — Upper bound on s

There is no (147, 227, 749)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 225 941479 233301 059132 435146 127905 835101 452576 030476 459655 010546 913141 > 2²²⁷ [i]