MinT - Best Known (227−69, 227, s)-Nets in Base 2

Best Known (227−69, 227, s)-Nets in Base 2

(227−69, 227, 112)-Net over F₂ — Constructive and digital

Digital (158, 227, 112)-net over F₂, using

23 times m-reduction [i] based on digital (158, 250, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 125, 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]

(227−69, 227, 193)-Net over F₂ — Digital

Digital (158, 227, 193)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(227−69, 227, 1306)-Net in Base 2 — Upper bound on s

There is no (158, 227, 1307)-net in base 2, because

1 times m-reduction [i] would yield (158, 226, 1307)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 108 096031 469268 870051 246539 120070 405121 174922 852884 077092 780341 294056 > 2²²⁶ [i]