MinT - Best Known (50−29, 50, s)-Nets in Base 2

Best Known (50−29, 50, s)-Nets in Base 2

(50−29, 50, 21)-Net over F₂ — Constructive and digital

Digital (21, 50, 21)-net over F₂, using

net from sequence [i] based on digital (21, 20)-sequence over F₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 21 and N(F) ≥ 21, using
  - an explicitly constructive algebraic function field [i]

(50−29, 50, 49)-Net over F₂ — Upper bound on s (digital)

There is no digital (21, 50, 50)-net over F₂, because

5 times m-reduction [i] would yield digital (21, 45, 50)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2⁴⁵, 50, F₂, 24) (dual of [50, 5, 25]-code), but
  - Griesmer bound [i]

(50−29, 50, 50)-Net in Base 2 — Upper bound on s

There is no (21, 50, 51)-net in base 2, because

1 times m-reduction [i] would yield (21, 49, 51)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 650 568189 312444 > 2⁴⁹ [i]