MinT - Best Known (138−55, 138, s)-Nets in Base 2

Best Known (138−55, 138, s)-Nets in Base 2

(138−55, 138, 60)-Net over F₂ — Constructive and digital

Digital (83, 138, 60)-net over F₂, using

2 times m-reduction [i] based on digital (83, 140, 60)-net over F₂, using
- trace code for nets [i] based on digital (13, 70, 30)-net over F₄, using
  - net from sequence [i] based on digital (13, 29)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 30, using
      - F₄ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(138−55, 138, 69)-Net over F₂ — Digital

Digital (83, 138, 69)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(138−55, 138, 329)-Net in Base 2 — Upper bound on s

There is no (83, 138, 330)-net in base 2, because

1 times m-reduction [i] would yield (83, 137, 330)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 175782 933101 598988 784402 467002 862767 361520 > 2¹³⁷ [i]