MinT - Best Known (165−61, 165, s)-Nets in Base 2

Best Known (165−61, 165, s)-Nets in Base 2

(165−61, 165, 68)-Net over F₂ — Constructive and digital

Digital (104, 165, 68)-net over F₂, using

1 times m-reduction [i] based on digital (104, 166, 68)-net over F₂, using
- trace code for nets [i] based on digital (21, 83, 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]

(165−61, 165, 94)-Net over F₂ — Digital

Digital (104, 165, 94)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(165−61, 165, 489)-Net in Base 2 — Upper bound on s

There is no (104, 165, 490)-net in base 2, because

1 times m-reduction [i] would yield (104, 164, 490)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 23 720924 029749 114739 504983 247435 810072 342952 881584 > 2¹⁶⁴ [i]