MinT - Best Known (251−85, 251, s)-Nets in Base 2

Best Known (251−85, 251, s)-Nets in Base 2

(251−85, 251, 112)-Net over F₂ — Constructive and digital

Digital (166, 251, 112)-net over F₂, using

t-expansion [i] based on digital (163, 251, 112)-net over F₂, using
- 9 times m-reduction [i] based on digital (163, 260, 112)-net over F₂, using
  - trace code for nets [i] based on digital (33, 130, 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]

(251−85, 251, 164)-Net over F₂ — Digital

Digital (166, 251, 164)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(251−85, 251, 961)-Net in Base 2 — Upper bound on s

There is no (166, 251, 962)-net in base 2, because

1 times m-reduction [i] would yield (166, 250, 962)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 1835 055113 784428 414497 829110 114738 905434 345471 121088 996048 863634 463719 148640 > 2²⁵⁰ [i]