MinT - Best Known (227−75, 227, s)-Nets in Base 2

Best Known (227−75, 227, s)-Nets in Base 2

(227−75, 227, 112)-Net over F₂ — Constructive and digital

Digital (152, 227, 112)-net over F₂, using

11 times m-reduction [i] based on digital (152, 238, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 119, 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]

(227−75, 227, 159)-Net over F₂ — Digital

Digital (152, 227, 159)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(227−75, 227, 957)-Net in Base 2 — Upper bound on s

There is no (152, 227, 958)-net in base 2, because

1 times m-reduction [i] would yield (152, 226, 958)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 111 610366 779059 308918 766621 804175 077909 081034 408434 218736 635076 544015 > 2²²⁶ [i]