MinT - Best Known (138, 138+41, s)-Nets in Base 2

Best Known (138, 138+41, s)-Nets in Base 2

(138, 138+41, 195)-Net over F₂ — Constructive and digital

Digital (138, 179, 195)-net over F₂, using

7 times m-reduction [i] based on digital (138, 186, 195)-net over F₂, using
- trace code for nets [i] based on digital (14, 62, 65)-net over F₈, using
  - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
      - the Suzuki function field over F₈ [i]

(138, 138+41, 321)-Net over F₂ — Digital

Digital (138, 179, 321)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(138, 138+41, 3937)-Net in Base 2 — Upper bound on s

There is no (138, 179, 3938)-net in base 2, because

1 times m-reduction [i] would yield (138, 178, 3938)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 383703 862449 234878 126543 261862 600979 381114 161372 295544 > 2¹⁷⁸ [i]