MinT - Best Known (188−59, 188, s)-Nets in Base 2

Best Known (188−59, 188, s)-Nets in Base 2

(188−59, 188, 112)-Net over F₂ — Constructive and digital

Digital (129, 188, 112)-net over F₂, using

4 times m-reduction [i] based on digital (129, 192, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 96, 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]

(188−59, 188, 152)-Net over F₂ — Digital

Digital (129, 188, 152)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(188−59, 188, 976)-Net in Base 2 — Upper bound on s

There is no (129, 188, 977)-net in base 2, because

1 times m-reduction [i] would yield (129, 187, 977)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 196 268570 435343 062540 112816 715043 079150 457034 243150 394208 > 2¹⁸⁷ [i]