MinT - Best Known (220−99, 220, s)-Nets in Base 2

Best Known (220−99, 220, s)-Nets in Base 2

(220−99, 220, 59)-Net over F₂ — Constructive and digital

Digital (121, 220, 59)-net over F₂, using

5 times m-reduction [i] based on digital (121, 225, 59)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 67, 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, 158, 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]

(220−99, 220, 80)-Net over F₂ — Digital

Digital (121, 220, 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]

(220−99, 220, 295)-Net in Base 2 — Upper bound on s

There is no (121, 220, 296)-net in base 2, because

1 times m-reduction [i] would yield (121, 219, 296)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²¹⁹, 296, S₂, 98), but
  - 4 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]