MinT - Best Known (138, 138+39, s)-Nets in Base 2

Best Known (138, 138+39, s)-Nets in Base 2

(138, 138+39, 196)-Net over F₂ — Constructive and digital

Digital (138, 177, 196)-net over F₂, using

2¹ times duplication [i] based on digital (137, 176, 196)-net over F₂, using
- trace code for nets [i] based on digital (5, 44, 49)-net over F₁₆, using
  - net from sequence [i] based on digital (5, 48)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 5 and N(F) ≥ 49, using
      - a function field by SÃ©mirat [i]

(138, 138+39, 354)-Net over F₂ — Digital

Digital (138, 177, 354)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(138, 138+39, 4844)-Net in Base 2 — Upper bound on s

There is no (138, 177, 4845)-net in base 2, because

1 times m-reduction [i] would yield (138, 176, 4845)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 96142 088000 297805 941446 004745 482456 652426 047042 060288 > 2¹⁷⁶ [i]