MinT - Best Known (231−78, 231, s)-Nets in Base 2

Best Known (231−78, 231, s)-Nets in Base 2

(231−78, 231, 112)-Net over F₂ — Constructive and digital

Digital (153, 231, 112)-net over F₂, using

9 times m-reduction [i] based on digital (153, 240, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 120, 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]

(231−78, 231, 153)-Net over F₂ — Digital

Digital (153, 231, 153)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(231−78, 231, 877)-Net in Base 2 — Upper bound on s

There is no (153, 231, 878)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3477 689996 806712 865316 502902 743693 034017 235626 850738 165179 840023 344032 > 2²³¹ [i]