MinT - Best Known (53−25, 53, s)-Nets in Base 2

Best Known (53−25, 53, s)-Nets in Base 2

(53−25, 53, 23)-Net over F₂ — Constructive and digital

Digital (28, 53, 23)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (7, 19, 11)-net over F₂, using
  - linear code embedding found in a computer search [i]
2. digital (9, 34, 12)-net over F₂, using
  - net from sequence [i] based on digital (9, 11)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 9 and N(F) ≥ 12, using
      - an explicitly constructive algebraic function field [i]

(53−25, 53, 25)-Net over F₂ — Digital

Digital (28, 53, 25)-net over F₂, using

net from sequence [i] based on digital (28, 24)-sequence over F₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 28 and N(F) ≥ 25, using
  - a curve from the manYPoints database [i]

(53−25, 53, 87)-Net in Base 2 — Upper bound on s

There is no (28, 53, 88)-net in base 2, because

1 times m-reduction [i] would yield (28, 52, 88)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁵², 88, S₂, 24), but
  - the linear programming bound shows that MÂ â‰¥ 2 587715 521886 450011 810360 721408 / 536 605819 945767 > 2⁵² [i]