MinT - Best Known (219−65, 219, s)-Nets in Base 2

Best Known (219−65, 219, s)-Nets in Base 2

(219−65, 219, 112)-Net over F₂ — Constructive and digital

Digital (154, 219, 112)-net over F₂, using

23 times m-reduction [i] based on digital (154, 242, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 121, 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]

(219−65, 219, 198)-Net over F₂ — Digital

Digital (154, 219, 198)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(219−65, 219, 1390)-Net in Base 2 — Upper bound on s

There is no (154, 219, 1391)-net in base 2, because

1 times m-reduction [i] would yield (154, 218, 1391)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 422790 173999 877866 690023 625769 454693 481850 597044 822934 203599 029795 > 2²¹⁸ [i]