MinT - Best Known (219−71, 219, s)-Nets in Base 2

Best Known (219−71, 219, s)-Nets in Base 2

(219−71, 219, 112)-Net over F₂ — Constructive and digital

Digital (148, 219, 112)-net over F₂, using

11 times m-reduction [i] based on digital (148, 230, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 115, 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]

(219−71, 219, 161)-Net over F₂ — Digital

Digital (148, 219, 161)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(219−71, 219, 992)-Net in Base 2 — Upper bound on s

There is no (148, 219, 993)-net in base 2, because

1 times m-reduction [i] would yield (148, 218, 993)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 434544 557583 102289 886130 361602 341847 714682 615152 719367 525314 099864 > 2²¹⁸ [i]