MinT - Best Known (50, 50+65, s)-Nets in Base 2

Best Known (50, 50+65, s)-Nets in Base 2

(50, 50+65, 35)-Net over F₂ — Constructive and digital

Digital (50, 115, 35)-net over F₂, using

t-expansion [i] based on digital (48, 115, 35)-net over F₂, using
- net from sequence [i] based on digital (48, 34)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 41, N(F)Â =Â 32, 1 place with degree 2, and 2 places with degree 4 [i] based on function field F/F₂ with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]

(50, 50+65, 40)-Net over F₂ — Digital

Digital (50, 115, 40)-net over F₂, using

net from sequence [i] based on digital (50, 39)-sequence over F₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 50 and N(F) ≥ 40, using
  - Drinfeld modules of rank 1 [i]

(50, 50+65, 109)-Net in Base 2 — Upper bound on s

There is no (50, 115, 110)-net in base 2, because

1 times m-reduction [i] would yield (50, 114, 110)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 23387 454093 238323 781683 119902 923963 > 2¹¹⁴ [i]