MinT - Best Known (135, 135+61, s)-Nets in Base 2

Best Known (135, 135+61, s)-Nets in Base 2

(135, 135+61, 112)-Net over F₂ — Constructive and digital

Digital (135, 196, 112)-net over F₂, using

8 times m-reduction [i] based on digital (135, 204, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 102, 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]

(135, 135+61, 161)-Net over F₂ — Digital

Digital (135, 196, 161)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(135, 135+61, 1046)-Net in Base 2 — Upper bound on s

There is no (135, 196, 1047)-net in base 2, because

1 times m-reduction [i] would yield (135, 195, 1047)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 50974 386725 216395 705641 162774 487224 750550 394882 944621 179008 > 2¹⁹⁵ [i]