MinT - Best Known (55−27, 55, s)-Nets in Base 2

Best Known (55−27, 55, s)-Nets in Base 2

(55−27, 55, 22)-Net over F₂ — Constructive and digital

Digital (28, 55, 22)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (7, 20, 11)-net over F₂, using
  - linear code embedding found in a computer search [i]
2. digital (8, 35, 11)-net over F₂, using
  - net from sequence [i] based on digital (8, 10)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 8 and N(F) ≥ 11, using
      - an explicitly constructive algebraic function field [i]

(55−27, 55, 25)-Net over F₂ — Digital

Digital (28, 55, 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]

(55−27, 55, 71)-Net over F₂ — Upper bound on s (digital)

There is no digital (28, 55, 72)-net over F₂, because

1 times m-reduction [i] would yield digital (28, 54, 72)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2⁵⁴, 72, F₂, 26) (dual of [72, 18, 27]-code), but
  - â€œJaâ€ bound on codes from Brouwerâ€™s database [i]

(55−27, 55, 75)-Net in Base 2 — Upper bound on s

There is no (28, 55, 76)-net in base 2, because

1 times m-reduction [i] would yield (28, 54, 76)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁵⁴, 76, S₂, 26), but
  - the linear programming bound shows that MÂ â‰¥ 15529 924724 648266 694656 / 830025 > 2⁵⁴ [i]