MinT - Best Known (138, 138+59, s)-Nets in Base 2

Best Known (138, 138+59, s)-Nets in Base 2

(138, 138+59, 112)-Net over F₂ — Constructive and digital

Digital (138, 197, 112)-net over F₂, using

13 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+59, 177)-Net over F₂ — Digital

Digital (138, 197, 177)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(138, 138+59, 1221)-Net in Base 2 — Upper bound on s

There is no (138, 197, 1222)-net in base 2, because

1 times m-reduction [i] would yield (138, 196, 1222)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 101752 951980 229917 653880 571315 006740 548692 587925 675200 143000 > 2¹⁹⁶ [i]