MinT - Best Known (81, 81+55, s)-Nets in Base 2

Best Known (81, 81+55, s)-Nets in Base 2

(81, 81+55, 60)-Net over F₂ — Constructive and digital

Digital (81, 136, 60)-net over F₂, using

trace code for nets [i] based on digital (13, 68, 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]

(81, 81+55, 66)-Net over F₂ — Digital

Digital (81, 136, 66)-net over F₂, using

trace code for nets [i] based on digital (13, 68, 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]

(81, 81+55, 311)-Net in Base 2 — Upper bound on s

There is no (81, 136, 312)-net in base 2, because

1 times m-reduction [i] would yield (81, 135, 312)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 45090 280780 103601 298208 353374 706167 437572 > 2¹³⁵ [i]