MinT - Best Known (212−65, 212, s)-Nets in Base 2

Best Known (212−65, 212, s)-Nets in Base 2

(212−65, 212, 112)-Net over F₂ — Constructive and digital

Digital (147, 212, 112)-net over F₂, using

16 times m-reduction [i] based on digital (147, 228, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 114, 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]

(212−65, 212, 178)-Net over F₂ — Digital

Digital (147, 212, 178)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(212−65, 212, 1188)-Net in Base 2 — Upper bound on s

There is no (147, 212, 1189)-net in base 2, because

1 times m-reduction [i] would yield (147, 211, 1189)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3317 702079 583553 928018 341189 139068 937323 817225 924474 884369 457029 > 2²¹¹ [i]