MinT - Best Known (232−77, 232, s)-Nets in Base 2

Best Known (232−77, 232, s)-Nets in Base 2

(232−77, 232, 112)-Net over F₂ — Constructive and digital

Digital (155, 232, 112)-net over F₂, using

12 times m-reduction [i] based on digital (155, 244, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 122, 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]

(232−77, 232, 160)-Net over F₂ — Digital

Digital (155, 232, 160)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(232−77, 232, 960)-Net in Base 2 — Upper bound on s

There is no (155, 232, 961)-net in base 2, because

1 times m-reduction [i] would yield (155, 231, 961)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3508 503131 402053 723128 214612 295546 811086 731750 570459 659587 775868 836960 > 2²³¹ [i]