MinT - Best Known (221−98, 221, s)-Nets in Base 2

Best Known (221−98, 221, s)-Nets in Base 2

(221−98, 221, 62)-Net over F₂ — Constructive and digital

Digital (123, 221, 62)-net over F₂, using

2 times m-reduction [i] based on digital (123, 223, 62)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (19, 69, 20)-net over F₂, using
    - net from sequence [i] based on digital (19, 19)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 19 and N(F) ≥ 20, 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]

(221−98, 221, 80)-Net over F₂ — Digital

Digital (123, 221, 80)-net over F₂, using

t-expansion [i] based on digital (121, 221, 80)-net over F₂, using
- net from sequence [i] based on digital (121, 79)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 121 and N(F) ≥ 80, using
    - Drinfeld modules of rank 1 [i]

(221−98, 221, 297)-Net in Base 2 — Upper bound on s

There is no (123, 221, 298)-net in base 2, because

extracting embedded orthogonal array [i] would yield OA(2²²¹, 298, S₂, 98), but
- 2 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]