MinT - Best Known (239−81, 239, s)-Nets in Base 2

Best Known (239−81, 239, s)-Nets in Base 2

(239−81, 239, 112)-Net over F₂ — Constructive and digital

Digital (158, 239, 112)-net over F₂, using

11 times m-reduction [i] based on digital (158, 250, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 125, 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]

(239−81, 239, 156)-Net over F₂ — Digital

Digital (158, 239, 156)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(239−81, 239, 916)-Net in Base 2 — Upper bound on s

There is no (158, 239, 917)-net in base 2, because

1 times m-reduction [i] would yield (158, 238, 917)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 443183 302113 988348 731267 036968 897399 342593 562507 430295 637229 128423 983944 > 2²³⁸ [i]