MinT - Best Known (125−49, 125, s)-Nets in Base 2

Best Known (125−49, 125, s)-Nets in Base 2

(125−49, 125, 60)-Net over F₂ — Constructive and digital

Digital (76, 125, 60)-net over F₂, using

1 times m-reduction [i] based on digital (76, 126, 60)-net over F₂, using
- trace code for nets [i] based on digital (13, 63, 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]

(125−49, 125, 66)-Net over F₂ — Digital

Digital (76, 125, 66)-net over F₂, using

1 times m-reduction [i] based on digital (76, 126, 66)-net over F₂, using
- trace code for nets [i] based on digital (13, 63, 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]

(125−49, 125, 318)-Net in Base 2 — Upper bound on s

There is no (76, 125, 319)-net in base 2, because

1 times m-reduction [i] would yield (76, 124, 319)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 22 522543 955408 141056 812464 161246 662149 > 2¹²⁴ [i]