MinT - Best Known (57, 57+31, s)-Nets in Base 2

Best Known (57, 57+31, s)-Nets in Base 2

(57, 57+31, 60)-Net over F₂ — Constructive and digital

Digital (57, 88, 60)-net over F₂, using

trace code for nets [i] based on digital (13, 44, 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]

(57, 57+31, 66)-Net over F₂ — Digital

Digital (57, 88, 66)-net over F₂, using

trace code for nets [i] based on digital (13, 44, 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]

(57, 57+31, 336)-Net in Base 2 — Upper bound on s

There is no (57, 88, 337)-net in base 2, because

1 times m-reduction [i] would yield (57, 87, 337)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 158 186160 844509 519200 483328 > 2⁸⁷ [i]