MinT - Best Known (235−111, 235, s)-Nets in Base 2

Best Known (235−111, 235, s)-Nets in Base 2

(235−111, 235, 59)-Net over F₂ — Constructive and digital

Digital (124, 235, 59)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (15, 70, 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, 165, 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]

(235−111, 235, 80)-Net over F₂ — Digital

Digital (124, 235, 80)-net over F₂, using

t-expansion [i] based on digital (121, 235, 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]

(235−111, 235, 267)-Net in Base 2 — Upper bound on s

There is no (124, 235, 268)-net in base 2, because

3 times m-reduction [i] would yield (124, 232, 268)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²³², 268, S₂, 108), but
  - the linear programming bound shows that MÂ â‰¥ 31 015917 364418 637531 270768 022329 527868 589587 272176 065660 880357 772788 956541 131711 053824 / 4226 244518 828475 > 2²³² [i]