MinT - Best Known (232−75, 232, s)-Nets in Base 2

Best Known (232−75, 232, s)-Nets in Base 2

(232−75, 232, 112)-Net over F₂ — Constructive and digital

Digital (157, 232, 112)-net over F₂, using

16 times m-reduction [i] based on digital (157, 248, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 124, 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]

(232−75, 232, 170)-Net over F₂ — Digital

Digital (157, 232, 170)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(232−75, 232, 1056)-Net in Base 2 — Upper bound on s

There is no (157, 232, 1057)-net in base 2, because

1 times m-reduction [i] would yield (157, 231, 1057)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3544 710679 441512 358942 359251 976118 087966 284500 866455 489485 998888 107296 > 2²³¹ [i]