MinT - Best Known (40−21, 40, s)-Nets in Base 2

Best Known (40−21, 40, s)-Nets in Base 2

(40−21, 40, 20)-Net over F₂ — Constructive and digital

Digital (19, 40, 20)-net over F₂, using

net from sequence [i] based on digital (19, 19)-sequence over F₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 19 and N(F) ≥ 20, using
  - an explicitly constructive algebraic function field [i]

(40−21, 40, 47)-Net over F₂ — Upper bound on s (digital)

There is no digital (19, 40, 48)-net over F₂, because

1 times m-reduction [i] would yield digital (19, 39, 48)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2³⁹, 48, F₂, 20) (dual of [48, 9, 21]-code), but
  - 1 times code embedding in larger space [i] would yield linear OA(2⁴⁰, 49, F₂, 20) (dual of [49, 9, 21]-code), but
    - â€œJaâ€ bound on codes from Brouwerâ€™s database [i]

(40−21, 40, 50)-Net in Base 2 — Upper bound on s

There is no (19, 40, 51)-net in base 2, because

1 times m-reduction [i] would yield (19, 39, 51)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2³⁹, 51, S₂, 20), but
  - the linear programming bound shows that MÂ â‰¥ 23503 160555 339776 / 40425 > 2³⁹ [i]