MinT - Best Known (220−101, 220, s)-Nets in Base 2

Best Known (220−101, 220, s)-Nets in Base 2

(220−101, 220, 59)-Net over F₂ — Constructive and digital

Digital (119, 220, 59)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (15, 65, 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, 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]

(220−101, 220, 73)-Net over F₂ — Digital

Digital (119, 220, 73)-net over F₂, using

t-expansion [i] based on digital (114, 220, 73)-net over F₂, using
- net from sequence [i] based on digital (114, 72)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 114 and N(F) ≥ 73, using
    - Drinfeld modules of rank 1 [i]

(220−101, 220, 264)-Net in Base 2 — Upper bound on s

There is no (119, 220, 265)-net in base 2, because

1 times m-reduction [i] would yield (119, 219, 265)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²¹⁹, 265, S₂, 100), but
  - the linear programming bound shows that MÂ â‰¥ 34611 787607 220460 636945 727454 704013 712621 341255 350789 664944 281364 723030 110196 727808 / 38195 169510 583125 > 2²¹⁹ [i]