MinT - Best Known (212−67, 212, s)-Nets in Base 2

Best Known (212−67, 212, s)-Nets in Base 2

(212−67, 212, 112)-Net over F₂ — Constructive and digital

Digital (145, 212, 112)-net over F₂, using

12 times m-reduction [i] based on digital (145, 224, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 112, 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]

(212−67, 212, 166)-Net over F₂ — Digital

Digital (145, 212, 166)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(212−67, 212, 1058)-Net in Base 2 — Upper bound on s

There is no (145, 212, 1059)-net in base 2, because

1 times m-reduction [i] would yield (145, 211, 1059)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3300 528560 516988 568863 832959 439994 936812 139296 547684 125724 308036 > 2²¹¹ [i]