MinT - Best Known (27, 52, s)-Nets in Base 2

Best Known (27, 52, s)-Nets in Base 2

(27, 52, 22)-Net over F₂ — Constructive and digital

Digital (27, 52, 22)-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 (8, 33, 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]

(27, 52, 24)-Net over F₂ — Digital

Digital (27, 52, 24)-net over F₂, using

t-expansion [i] based on digital (25, 52, 24)-net over F₂, using
- net from sequence [i] based on digital (25, 23)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 25 and N(F) ≥ 24, using
    - Drinfeld modules of rank 1 [i]

(27, 52, 78)-Net over F₂ — Upper bound on s (digital)

There is no digital (27, 52, 79)-net over F₂, because

1 times m-reduction [i] would yield digital (27, 51, 79)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2⁵¹, 79, F₂, 24) (dual of [79, 28, 25]-code), but
  - â€œJaâ€ bound on codes from Brouwerâ€™s database [i]

(27, 52, 80)-Net in Base 2 — Upper bound on s

There is no (27, 52, 81)-net in base 2, because

1 times m-reduction [i] would yield (27, 51, 81)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁵¹, 81, S₂, 24), but
  - the linear programming bound shows that MÂ â‰¥ 9 782701 960866 810384 351232 / 4249 786905 > 2⁵¹ [i]