MinT - Best Known (158−45, 158, s)-Nets in Base 2

Best Known (158−45, 158, s)-Nets in Base 2

(158−45, 158, 112)-Net over F₂ — Constructive and digital

Digital (113, 158, 112)-net over F₂, using

2 times m-reduction [i] based on digital (113, 160, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 80, 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]

(158−45, 158, 166)-Net over F₂ — Digital

Digital (113, 158, 166)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(158−45, 158, 1241)-Net in Base 2 — Upper bound on s

There is no (113, 158, 1242)-net in base 2, because

1 times m-reduction [i] would yield (113, 157, 1242)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 183617 639949 968918 597212 406723 906596 796623 739356 > 2¹⁵⁷ [i]