MinT - Best Known (119−37, 119, s)-Nets in Base 2

Best Known (119−37, 119, s)-Nets in Base 2

(119−37, 119, 68)-Net over F₂ — Constructive and digital

Digital (82, 119, 68)-net over F₂, using

3 times m-reduction [i] based on digital (82, 122, 68)-net over F₂, using
- trace code for nets [i] based on digital (21, 61, 34)-net over F₄, using
  - net from sequence [i] based on digital (21, 33)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
      - T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(119−37, 119, 108)-Net over F₂ — Digital

Digital (82, 119, 108)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(119−37, 119, 684)-Net in Base 2 — Upper bound on s

There is no (82, 119, 685)-net in base 2, because

1 times m-reduction [i] would yield (82, 118, 685)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 337265 829022 453602 646695 812710 198460 > 2¹¹⁸ [i]