MinT - Best Known (223−98, 223, s)-Nets in Base 2

Best Known (223−98, 223, s)-Nets in Base 2

(223−98, 223, 63)-Net over F₂ — Constructive and digital

Digital (125, 223, 63)-net over F₂, using

2 times m-reduction [i] based on digital (125, 225, 63)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 71, 21)-net over F₂, using
    - net from sequence [i] based on digital (21, 20)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 21 and N(F) ≥ 21, 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]

(223−98, 223, 82)-Net over F₂ — Digital

Digital (125, 223, 82)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(223−98, 223, 299)-Net in Base 2 — Upper bound on s

There is no (125, 223, 300)-net in base 2, because

extracting embedded orthogonal array [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]