MinT - Best Known (136, 136+57, s)-Nets in Base 2

Best Known (136, 136+57, s)-Nets in Base 2

(136, 136+57, 112)-Net over F₂ — Constructive and digital

Digital (136, 193, 112)-net over F₂, using

13 times m-reduction [i] based on digital (136, 206, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 103, 56)-net over F₄, using
  - net from sequence [i] based on digital (33, 55)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
      - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(136, 136+57, 180)-Net over F₂ — Digital

Digital (136, 193, 180)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(136, 136+57, 1268)-Net in Base 2 — Upper bound on s

There is no (136, 193, 1269)-net in base 2, because

1 times m-reduction [i] would yield (136, 192, 1269)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 6289 429490 035075 831139 907212 579967 747880 332050 835852 035916 > 2¹⁹² [i]