MinT - Best Known (138−33, 138, s)-Nets in Base 2

Best Known (138−33, 138, s)-Nets in Base 2

(138−33, 138, 144)-Net over F₂ — Constructive and digital

Digital (105, 138, 144)-net over F₂, using

3 times m-reduction [i] based on digital (105, 141, 144)-net over F₂, using
- trace code for nets [i] based on digital (11, 47, 48)-net over F₈, using
  - net from sequence [i] based on digital (11, 47)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 11 and N(F) ≥ 48, using
      - a function field by SÃ©mirat [i]

(138−33, 138, 232)-Net over F₂ — Digital

Digital (105, 138, 232)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(138−33, 138, 2547)-Net in Base 2 — Upper bound on s

There is no (105, 138, 2548)-net in base 2, because

1 times m-reduction [i] would yield (105, 137, 2548)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 174623 564001 899845 080452 825152 303784 187099 > 2¹³⁷ [i]