MinT - Best Known (210−93, 210, s)-Nets in Base 2

Best Known (210−93, 210, s)-Nets in Base 2

(210−93, 210, 59)-Net over F₂ — Constructive and digital

Digital (117, 210, 59)-net over F₂, using

3 times m-reduction [i] based on digital (117, 213, 59)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 63, 17)-net over F₂, using
    - net from sequence [i] based on digital (15, 16)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 15 and N(F) ≥ 17, using
        an explicitly constructive algebraic function field [i]
  2. digital (54, 150, 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]

(210−93, 210, 76)-Net over F₂ — Digital

Digital (117, 210, 76)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(210−93, 210, 296)-Net in Base 2 — Upper bound on s

There is no (117, 210, 297)-net in base 2, because

1 times m-reduction [i] would yield (117, 209, 297)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²⁰⁹, 297, S₂, 92), but
  - 3 times code embedding in larger space [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]