MinT - Best Known (231−81, 231, s)-Nets in Base 2

Best Known (231−81, 231, s)-Nets in Base 2

(231−81, 231, 112)-Net over F₂ — Constructive and digital

Digital (150, 231, 112)-net over F₂, using

3 times m-reduction [i] based on digital (150, 234, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 117, 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]

(231−81, 231, 141)-Net over F₂ — Digital

Digital (150, 231, 141)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(231−81, 231, 790)-Net in Base 2 — Upper bound on s

There is no (150, 231, 791)-net in base 2, because

1 times m-reduction [i] would yield (150, 230, 791)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 1728 651397 329717 832431 238389 684761 320690 376401 455897 615241 512807 271824 > 2²³⁰ [i]