MinT - Best Known (228−69, 228, s)-Nets in Base 2

Best Known (228−69, 228, s)-Nets in Base 2

(228−69, 228, 112)-Net over F₂ — Constructive and digital

Digital (159, 228, 112)-net over F₂, using

24 times m-reduction [i] based on digital (159, 252, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 126, 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]

(228−69, 228, 195)-Net over F₂ — Digital

Digital (159, 228, 195)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(228−69, 228, 1334)-Net in Base 2 — Upper bound on s

There is no (159, 228, 1335)-net in base 2, because

1 times m-reduction [i] would yield (159, 227, 1335)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 216 591328 519264 377018 660825 031869 173971 506194 240765 569878 759925 130685 > 2²²⁷ [i]