MinT - Best Known (67, 67+41, s)-Nets in Base 2

Best Known (67, 67+41, s)-Nets in Base 2

(67, 67+41, 60)-Net over F₂ — Constructive and digital

Digital (67, 108, 60)-net over F₂, using

trace code for nets [i] based on digital (13, 54, 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]

(67, 67+41, 66)-Net over F₂ — Digital

Digital (67, 108, 66)-net over F₂, using

trace code for nets [i] based on digital (13, 54, 33)-net over F₄, using
- net from sequence [i] based on digital (13, 32)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 33, using
    - Drinfeld modules of rank 1 [i]

(67, 67+41, 310)-Net in Base 2 — Upper bound on s

There is no (67, 108, 311)-net in base 2, because

1 times m-reduction [i] would yield (67, 107, 311)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 168 950044 415498 668280 469230 622649 > 2¹⁰⁷ [i]