MinT - Best Known (231−73, 231, s)-Nets in Base 2

Best Known (231−73, 231, s)-Nets in Base 2

(231−73, 231, 112)-Net over F₂ — Constructive and digital

Digital (158, 231, 112)-net over F₂, using

19 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]

(231−73, 231, 178)-Net over F₂ — Digital

Digital (158, 231, 178)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(231−73, 231, 1144)-Net in Base 2 — Upper bound on s

There is no (158, 231, 1145)-net in base 2, because

1 times m-reduction [i] would yield (158, 230, 1145)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 1763 950071 372043 552118 953251 575916 988774 735904 714059 437234 195345 685098 > 2²³⁰ [i]