MinT - Best Known (178−51, 178, s)-Nets in Base 2

Best Known (178−51, 178, s)-Nets in Base 2

(178−51, 178, 112)-Net over F₂ — Constructive and digital

Digital (127, 178, 112)-net over F₂, using

10 times m-reduction [i] based on digital (127, 188, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 94, 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]

(178−51, 178, 181)-Net over F₂ — Digital

Digital (127, 178, 181)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(178−51, 178, 1340)-Net in Base 2 — Upper bound on s

There is no (127, 178, 1341)-net in base 2, because

1 times m-reduction [i] would yield (127, 177, 1341)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 193668 737995 726912 754937 715556 135147 028380 599064 660870 > 2¹⁷⁷ [i]