MinT - Best Known (109, 109+65, s)-Nets in Base 2

Best Known (109, 109+65, s)-Nets in Base 2

(109, 109+65, 68)-Net over F₂ — Constructive and digital

Digital (109, 174, 68)-net over F₂, using

2 times m-reduction [i] based on digital (109, 176, 68)-net over F₂, using
- trace code for nets [i] based on digital (21, 88, 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]

(109, 109+65, 96)-Net over F₂ — Digital

Digital (109, 174, 96)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(109, 109+65, 496)-Net in Base 2 — Upper bound on s

There is no (109, 174, 497)-net in base 2, because

1 times m-reduction [i] would yield (109, 173, 497)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 12172 950783 852973 450435 101198 426726 969106 847838 157493 > 2¹⁷³ [i]