MinT - Best Known (216−97, 216, s)-Nets in Base 2

Best Known (216−97, 216, s)-Nets in Base 2

(216−97, 216, 59)-Net over F₂ — Constructive and digital

Digital (119, 216, 59)-net over F₂, using

3 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]

(216−97, 216, 76)-Net over F₂ — Digital

Digital (119, 216, 76)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(216−97, 216, 294)-Net in Base 2 — Upper bound on s

There is no (119, 216, 295)-net in base 2, because

1 times m-reduction [i] would yield (119, 215, 295)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²¹⁵, 295, S₂, 96), but
  - 4 times code embedding in larger space [i] would yield OA(2²¹⁹, 299, S₂, 96), but
    - adding a parity check bit [i] would yield OA(2²²⁰, 300, S₂, 97), but
      - the linear programming bound shows that MÂ â‰¥ 2 707506 179671 220537 436761 186277 555172 982275 381427 765331 317661 637213 928123 767152 572244 071765 637582 553088 / 1 434637 915544 533146 412054 075529 090625 > 2²²⁰ [i]