MinT - Best Known (69−21, 69, s)-Nets in Base 2

Best Known (69−21, 69, s)-Nets in Base 2

(69−21, 69, 60)-Net over F₂ — Constructive and digital

Digital (48, 69, 60)-net over F₂, using

1 times m-reduction [i] based on digital (48, 70, 60)-net over F₂, using
- trace code for nets [i] based on digital (13, 35, 30)-net over F₄, using
  - net from sequence [i] based on digital (13, 29)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 30, using
      - F₄ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(69−21, 69, 77)-Net over F₂ — Digital

Digital (48, 69, 77)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(69−21, 69, 490)-Net in Base 2 — Upper bound on s

There is no (48, 69, 491)-net in base 2, because

1 times m-reduction [i] would yield (48, 68, 491)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 299 138021 506810 811649 > 2⁶⁸ [i]