MinT - Best Known (212−95, 212, s)-Nets in Base 2

Best Known (212−95, 212, s)-Nets in Base 2

(212−95, 212, 59)-Net over F₂ — Constructive and digital

Digital (117, 212, 59)-net over F₂, using

1 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]

(212−95, 212, 75)-Net over F₂ — Digital

Digital (117, 212, 75)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(212−95, 212, 294)-Net in Base 2 — Upper bound on s

There is no (117, 212, 295)-net in base 2, because

1 times m-reduction [i] would yield (117, 211, 295)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²¹¹, 295, S₂, 94), but
  - 4 times code embedding in larger space [i] would yield OA(2²¹⁵, 299, S₂, 94), but
    - adding a parity check bit [i] would yield OA(2²¹⁶, 300, S₂, 95), but
      - the linear programming bound shows that MÂ â‰¥ 872407 680551 113098 866799 953294 886066 311827 924058 628412 428016 149220 115296 187426 023936 653562 740736 / 5 938078 052347 021578 779474 608305 > 2²¹⁶ [i]