MinT - Best Known (55−26, 55, s)-Nets in Base 2

Best Known (55−26, 55, s)-Nets in Base 2

(55−26, 55, 23)-Net over F₂ — Constructive and digital

Digital (29, 55, 23)-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 (9, 35, 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]

(55−26, 55, 25)-Net over F₂ — Digital

Digital (29, 55, 25)-net over F₂, using

t-expansion [i] based on 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]

(55−26, 55, 79)-Net over F₂ — Upper bound on s (digital)

There is no digital (29, 55, 80)-net over F₂, because

extracting embedded orthogonal array [i] would yield linear OA(2⁵⁵, 80, F₂, 26) (dual of [80, 25, 27]-code), but
- â€œJaâ€ bound on codes from Brouwerâ€™s database [i]

(55−26, 55, 81)-Net in Base 2 — Upper bound on s

There is no (29, 55, 82)-net in base 2, because

extracting embedded orthogonal array [i] would yield OA(2⁵⁵, 82, S₂, 26), but
- the linear programming bound shows that MÂ â‰¥ 644 047496 602240 790095 200256 / 16885 995399 > 2⁵⁵ [i]