MinT - Best Known (128−51, 128, s)-Nets in Base 2

Best Known (128−51, 128, s)-Nets in Base 2

(128−51, 128, 60)-Net over F₂ — Constructive and digital

Digital (77, 128, 60)-net over F₂, using

trace code for nets [i] based on digital (13, 64, 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]

(128−51, 128, 66)-Net over F₂ — Digital

Digital (77, 128, 66)-net over F₂, using

trace code for nets [i] based on digital (13, 64, 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]

(128−51, 128, 308)-Net in Base 2 — Upper bound on s

There is no (77, 128, 309)-net in base 2, because

1 times m-reduction [i] would yield (77, 127, 309)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 170 755458 295472 547085 256391 969405 352184 > 2¹²⁷ [i]