MinT - Best Known (129−41, 129, s)-Nets in Base 2

Best Known (129−41, 129, s)-Nets in Base 2

(129−41, 129, 68)-Net over F₂ — Constructive and digital

Digital (88, 129, 68)-net over F₂, using

5 times m-reduction [i] based on digital (88, 134, 68)-net over F₂, using
- trace code for nets [i] based on digital (21, 67, 34)-net over F₄, using
  - net from sequence [i] based on digital (21, 33)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
      - T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(129−41, 129, 109)-Net over F₂ — Digital

Digital (88, 129, 109)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(129−41, 129, 672)-Net in Base 2 — Upper bound on s

There is no (88, 129, 673)-net in base 2, because

1 times m-reduction [i] would yield (88, 128, 673)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 346 719284 056149 367034 485213 412223 156616 > 2¹²⁸ [i]