MinT - Best Known (40, 57, s)-Nets in Base 2

Best Known (40, 57, s)-Nets in Base 2

(40, 57, 55)-Net over F₂ — Constructive and digital

Digital (40, 57, 55)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (5, 13, 11)-net over F₂, using
  - linear code embedding found in a computer search [i]
2. digital (27, 44, 44)-net over F₂, using
  - a shift-net [i]

(40, 57, 71)-Net over F₂ — Digital

Digital (40, 57, 71)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(40, 57, 470)-Net in Base 2 — Upper bound on s

There is no (40, 57, 471)-net in base 2, because

1 times m-reduction [i] would yield (40, 56, 471)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 72709 806769 910818 > 2⁵⁶ [i]