MinT - Best Known (129−51, 129, s)-Nets in Base 2

Best Known (129−51, 129, s)-Nets in Base 2

(129−51, 129, 60)-Net over F₂ — Constructive and digital

Digital (78, 129, 60)-net over F₂, using

1 times m-reduction [i] based on digital (78, 130, 60)-net over F₂, using
- trace code for nets [i] based on digital (13, 65, 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]

(129−51, 129, 66)-Net over F₂ — Digital

Digital (78, 129, 66)-net over F₂, using

1 times m-reduction [i] based on digital (78, 130, 66)-net over F₂, using
- trace code for nets [i] based on digital (13, 65, 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]

(129−51, 129, 318)-Net in Base 2 — Upper bound on s

There is no (78, 129, 319)-net in base 2, because

1 times m-reduction [i] would yield (78, 128, 319)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 350 466464 490006 704399 021839 286361 439624 > 2¹²⁸ [i]