MinT - Best Known (212−92, 212, s)-Nets in Base 2

Best Known (212−92, 212, s)-Nets in Base 2

(212−92, 212, 62)-Net over F₂ — Constructive and digital

Digital (120, 212, 62)-net over F₂, using

2 times m-reduction [i] based on digital (120, 214, 62)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (19, 66, 20)-net over F₂, using
    - net from sequence [i] based on digital (19, 19)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 19 and N(F) ≥ 20, using
        an explicitly constructive algebraic function field [i]
  2. digital (54, 148, 42)-net over F₂, using
    - net from sequence [i] based on digital (54, 41)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using
        an explicitly constructive algebraic function field [i]

(212−92, 212, 80)-Net over F₂ — Digital

Digital (120, 212, 80)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(212−92, 212, 299)-Net in Base 2 — Upper bound on s

There is no (120, 212, 300)-net in base 2, because

extracting embedded orthogonal array [i] would yield OA(2²¹², 300, S₂, 92), but
- the linear programming bound shows that MÂ â‰¥ 2484 132694 351399 471861 447697 940992 402712 231248 004226 234322 576635 442083 069406 167011 327832 607273 517056 / 377277 818302 855799 777736 052112 623275 > 2²¹² [i]