MinT - Best Known (68, 68+187, s)-Nets in Base 2

Best Known (68, 68+187, s)-Nets in Base 2

(68, 68+187, 43)-Net over F₂ — Constructive and digital

Digital (68, 255, 43)-net over F₂, using

t-expansion [i] based on digital (59, 255, 43)-net over F₂, using
- net from sequence [i] based on digital (59, 42)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 54, N(F)Â =Â 42, and 1 place with degree 6 [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]

(68, 68+187, 49)-Net over F₂ — Digital

Digital (68, 255, 49)-net over F₂, using

net from sequence [i] based on digital (68, 48)-sequence over F₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 68 and N(F) ≥ 49, using
  - an algebraic function field by Auer [i]

(68, 68+187, 86)-Net in Base 2 — Upper bound on s

There is no (68, 255, 87)-net in base 2, because

2 times m-reduction [i] would yield (68, 253, 87)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2²⁵³, 87, S₂, 3, 185), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 694752 535423 897172 541425 910052 127447 119619 907993 843384 236745 504047 478777 839616 / 31 > 2²⁵³ [i]