MinT - Best Known (109, 109+63, s)-Nets in Base 2

Best Known (109, 109+63, s)-Nets in Base 2

(109, 109+63, 68)-Net over F₂ — Constructive and digital

Digital (109, 172, 68)-net over F₂, using

4 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+63, 99)-Net over F₂ — Digital

Digital (109, 172, 99)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(109, 109+63, 523)-Net in Base 2 — Upper bound on s

There is no (109, 172, 524)-net in base 2, because

1 times m-reduction [i] would yield (109, 171, 524)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3009 946441 534222 473999 747613 966554 821932 435804 717440 > 2¹⁷¹ [i]