MinT - Best Known (165−47, 165, s)-Nets in Base 2

Best Known (165−47, 165, s)-Nets in Base 2

(165−47, 165, 112)-Net over F₂ — Constructive and digital

Digital (118, 165, 112)-net over F₂, using

5 times m-reduction [i] based on digital (118, 170, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 85, 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]

(165−47, 165, 172)-Net over F₂ — Digital

Digital (118, 165, 172)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(165−47, 165, 1287)-Net in Base 2 — Upper bound on s

There is no (118, 165, 1288)-net in base 2, because

1 times m-reduction [i] would yield (118, 164, 1288)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 23 596890 834081 027246 491446 765462 911026 137525 712912 > 2¹⁶⁴ [i]