MinT - Best Known (138, 138+57, s)-Nets in Base 2

Best Known (138, 138+57, s)-Nets in Base 2

(138, 138+57, 112)-Net over F₂ — Constructive and digital

Digital (138, 195, 112)-net over F₂, using

15 times m-reduction [i] based on digital (138, 210, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 105, 56)-net over F₄, using
  - net from sequence [i] based on digital (33, 55)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
      - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(138, 138+57, 186)-Net over F₂ — Digital

Digital (138, 195, 186)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(138, 138+57, 1335)-Net in Base 2 — Upper bound on s

There is no (138, 195, 1336)-net in base 2, because

1 times m-reduction [i] would yield (138, 194, 1336)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 25436 421913 324489 759977 983907 723563 778173 280787 611888 767580 > 2¹⁹⁴ [i]