MinT - Best Known (155−45, 155, s)-Nets in Base 2

Best Known (155−45, 155, s)-Nets in Base 2

(155−45, 155, 84)-Net over F₂ — Constructive and digital

Digital (110, 155, 84)-net over F₂, using

t-expansion [i] based on digital (109, 155, 84)-net over F₂, using
- 1 times m-reduction [i] based on 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]

(155−45, 155, 86)-Net in Base 2 — Constructive

(110, 155, 86)-net in base 2, using

5 times m-reduction [i] based on (110, 160, 86)-net in base 2, using
- trace code for nets [i] based on (30, 80, 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]

(155−45, 155, 156)-Net over F₂ — Digital

Digital (110, 155, 156)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(155−45, 155, 1126)-Net in Base 2 — Upper bound on s

There is no (110, 155, 1127)-net in base 2, because

1 times m-reduction [i] would yield (110, 154, 1127)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 22899 027095 903231 864943 830780 196986 685621 594498 > 2¹⁵⁴ [i]