MinT - Best Known (50, 50+206, s)-Nets in Base 2

Best Known (50, 50+206, s)-Nets in Base 2

(50, 50+206, 35)-Net over F₂ — Constructive and digital

Digital (50, 256, 35)-net over F₂, using

t-expansion [i] based on digital (48, 256, 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+206, 40)-Net over F₂ — Digital

Digital (50, 256, 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+206, 61)-Net in Base 2 — Upper bound on s

There is no (50, 256, 62)-net in base 2, because

16 times m-reduction [i] would yield (50, 240, 62)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2²⁴⁰, 62, S₂, 4, 190), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 508 851954 656174 686919 989680 213960 532558 315362 300178 259939 022585 972274 495488 / 191 > 2²⁴⁰ [i]