MinT - Best Known (219−74, 219, s)-Nets in Base 2

Best Known (219−74, 219, s)-Nets in Base 2

(219−74, 219, 112)-Net over F₂ — Constructive and digital

Digital (145, 219, 112)-net over F₂, using

5 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]

(219−74, 219, 146)-Net over F₂ — Digital

Digital (145, 219, 146)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(219−74, 219, 833)-Net in Base 2 — Upper bound on s

There is no (145, 219, 834)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 877652 522922 172273 104411 830351 651974 076049 107045 111835 113829 180560 > 2²¹⁹ [i]