MinT - Best Known (123−47, 123, s)-Nets in Base 2

Best Known (123−47, 123, s)-Nets in Base 2

(123−47, 123, 60)-Net over F₂ — Constructive and digital

Digital (76, 123, 60)-net over F₂, using

3 times m-reduction [i] based on digital (76, 126, 60)-net over F₂, using
- trace code for nets [i] based on digital (13, 63, 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]

(123−47, 123, 69)-Net over F₂ — Digital

Digital (76, 123, 69)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(123−47, 123, 339)-Net in Base 2 — Upper bound on s

There is no (76, 123, 340)-net in base 2, because

1 times m-reduction [i] would yield (76, 122, 340)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 5 339782 256869 192265 782114 265359 803584 > 2¹²² [i]