MinT - Best Known (50, 50+140, s)-Nets in Base 2

Best Known (50, 50+140, s)-Nets in Base 2

(50, 50+140, 35)-Net over F₂ — Constructive and digital

Digital (50, 190, 35)-net over F₂, using

t-expansion [i] based on digital (48, 190, 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+140, 40)-Net over F₂ — Digital

Digital (50, 190, 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+140, 65)-Net in Base 2 — Upper bound on s

There is no (50, 190, 66)-net in base 2, because

1 times m-reduction [i] would yield (50, 189, 66)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2¹⁸⁹, 66, S₂, 3, 139), but
  - the LP bound with quadratic polynomials shows that MÂ â‰¥ 29816 233243 086733 628219 999760 236415 476486 188361 204163 936256 / 35 > 2¹⁸⁹ [i]