MinT - Best Known (157, 157+77, s)-Nets in Base 2

Best Known (157, 157+77, s)-Nets in Base 2

(157, 157+77, 112)-Net over F₂ — Constructive and digital

Digital (157, 234, 112)-net over F₂, using

14 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]

(157, 157+77, 165)-Net over F₂ — Digital

Digital (157, 234, 165)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(157, 157+77, 998)-Net in Base 2 — Upper bound on s

There is no (157, 234, 999)-net in base 2, because

1 times m-reduction [i] would yield (157, 233, 999)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 14184 325746 924474 649598 996646 845565 030927 876249 863336 756530 271554 089383 > 2²³³ [i]