MinT - Best Known (111, 111+45, s)-Nets in Base 2

Best Known (111, 111+45, s)-Nets in Base 2

(111, 111+45, 112)-Net over F₂ — Constructive and digital

Digital (111, 156, 112)-net over F₂, using

trace code for nets [i] based on digital (33, 78, 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]

(111, 111+45, 159)-Net over F₂ — Digital

Digital (111, 156, 159)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(111, 111+45, 1163)-Net in Base 2 — Upper bound on s

There is no (111, 156, 1164)-net in base 2, because

1 times m-reduction [i] would yield (111, 155, 1164)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 45723 961560 552662 451215 144828 325423 886522 474256 > 2¹⁵⁵ [i]