MinT - Best Known (251−89, 251, s)-Nets in Base 2

Best Known (251−89, 251, s)-Nets in Base 2

(251−89, 251, 112)-Net over F₂ — Constructive and digital

Digital (162, 251, 112)-net over F₂, using

7 times m-reduction [i] based on digital (162, 258, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 129, 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]

(251−89, 251, 147)-Net over F₂ — Digital

Digital (162, 251, 147)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(251−89, 251, 822)-Net in Base 2 — Upper bound on s

There is no (162, 251, 823)-net in base 2, because

1 times m-reduction [i] would yield (162, 250, 823)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 1863 939713 975553 033638 453301 558302 615675 893981 013556 509074 990594 881131 886066 > 2²⁵⁰ [i]