MinT - Best Known (109, 156, s)-Nets in Base 2

Best Known (109, 156, s)-Nets in Base 2

(109, 156, 84)-Net over F₂ — Constructive and digital

Digital (109, 156, 84)-net over F₂, using

trace code for nets [i] based on digital (5, 52, 28)-net over F₈, using
- net from sequence [i] based on digital (5, 27)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 5 and N(F) ≥ 28, using
    - a function field by SÃ©mirat [i]

(109, 156, 86)-Net in Base 2 — Constructive

(109, 156, 86)-net in base 2, using

2 times m-reduction [i] based on (109, 158, 86)-net in base 2, using
- trace code for nets [i] based on (30, 79, 43)-net in base 4, using
  - net from sequence [i] based on (30, 42)-sequence in base 4, using
    - base expansion [i] based on digital (60, 42)-sequence over F₂, using
      - t-expansion [i] based on digital (59, 42)-sequence over F₂, using
        Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 54, N(F)Â =Â 42, and 1 place with degree 6 [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]

(109, 156, 144)-Net over F₂ — Digital

Digital (109, 156, 144)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(109, 156, 973)-Net in Base 2 — Upper bound on s

There is no (109, 156, 974)-net in base 2, because

1 times m-reduction [i] would yield (109, 155, 974)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 45920 180985 604483 437588 067167 249702 725287 737568 > 2¹⁵⁵ [i]