MinT - Best Known (224−101, 224, s)-Nets in Base 2

Best Known (224−101, 224, s)-Nets in Base 2

(224−101, 224, 62)-Net over F₂ — Constructive and digital

Digital (123, 224, 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, 155, 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]

(224−101, 224, 80)-Net over F₂ — Digital

Digital (123, 224, 80)-net over F₂, using

t-expansion [i] based on digital (121, 224, 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]

(224−101, 224, 280)-Net in Base 2 — Upper bound on s

There is no (123, 224, 281)-net in base 2, because

1 times m-reduction [i] would yield (123, 223, 281)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²²³, 281, S₂, 100), but
  - the linear programming bound shows that MÂ â‰¥ 6 731852 950873 970513 872016 536259 901060 971832 416172 423981 783268 435176 251291 176198 153718 202368 / 423281 780840 464014 331863 > 2²²³ [i]