MinT - Best Known (213−94, 213, s)-Nets in Base 2

Best Known (213−94, 213, s)-Nets in Base 2

(213−94, 213, 59)-Net over F₂ — Constructive and digital

Digital (119, 213, 59)-net over F₂, using

6 times m-reduction [i] based on digital (119, 219, 59)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 65, 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, 154, 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]

(213−94, 213, 78)-Net over F₂ — Digital

Digital (119, 213, 78)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(213−94, 213, 296)-Net in Base 2 — Upper bound on s

There is no (119, 213, 297)-net in base 2, because

extracting embedded orthogonal array [i] would yield OA(2²¹³, 297, S₂, 94), but
- 2 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]