MinT - Best Known (226−79, 226, s)-Nets in Base 2

Best Known (226−79, 226, s)-Nets in Base 2

(226−79, 226, 112)-Net over F₂ — Constructive and digital

Digital (147, 226, 112)-net over F₂, using

2 times m-reduction [i] based on digital (147, 228, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 114, 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]

(226−79, 226, 139)-Net over F₂ — Digital

Digital (147, 226, 139)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(226−79, 226, 783)-Net in Base 2 — Upper bound on s

There is no (147, 226, 784)-net in base 2, because

1 times m-reduction [i] would yield (147, 225, 784)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 55 209425 140765 058775 528682 880628 530913 060639 850567 508354 623648 296808 > 2²²⁵ [i]