MinT - Best Known (237−77, 237, s)-Nets in Base 2

Best Known (237−77, 237, s)-Nets in Base 2

(237−77, 237, 112)-Net over F₂ — Constructive and digital

Digital (160, 237, 112)-net over F₂, using

17 times m-reduction [i] based on digital (160, 254, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 127, 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]

(237−77, 237, 171)-Net over F₂ — Digital

Digital (160, 237, 171)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(237−77, 237, 1057)-Net in Base 2 — Upper bound on s

There is no (160, 237, 1058)-net in base 2, because

1 times m-reduction [i] would yield (160, 236, 1058)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 112612 013733 970505 789557 396525 551581 011429 083668 166415 965155 442892 603136 > 2²³⁶ [i]