MinT - Best Known (216−98, 216, s)-Nets in Base 2

Best Known (216−98, 216, s)-Nets in Base 2

(216−98, 216, 59)-Net over F₂ — Constructive and digital

Digital (118, 216, 59)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (15, 64, 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, 152, 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]

(216−98, 216, 74)-Net over F₂ — Digital

Digital (118, 216, 74)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(216−98, 216, 292)-Net in Base 2 — Upper bound on s

There is no (118, 216, 293)-net in base 2, because

extracting embedded orthogonal array [i] would yield OA(2²¹⁶, 293, S₂, 98), but
- 7 times code embedding in larger space [i] would yield OA(2²²³, 300, S₂, 98), but
  - the linear programming bound shows that MÂ â‰¥ 1 797227 776840 883681 896516 291018 079602 556004 736527 384481 376471 307374 972535 986485 205069 933364 379648 / 112622 624143 760877 566730 935625 > 2²²³ [i]