MinT - Best Known (56−48, 56, s)-Nets in Base 3

Best Known (56−48, 56, s)-Nets in Base 3

(56−48, 56, 16)-Net over F₃ — Constructive and digital

Digital (8, 56, 16)-net over F₃, using

t-expansion [i] based on digital (7, 56, 16)-net over F₃, using
- net from sequence [i] based on digital (7, 15)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 7 and N(F) ≥ 16, using
    - an explicitly constructive algebraic function field [i]

(56−48, 56, 17)-Net over F₃ — Digital

Digital (8, 56, 17)-net over F₃, using

net from sequence [i] based on digital (8, 16)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 8 and N(F) ≥ 17, using
  - a curve from the manYPoints database [i]

(56−48, 56, 24)-Net in Base 3 — Upper bound on s

There is no (8, 56, 25)-net in base 3, because

16 times m-reduction [i] would yield (8, 40, 25)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3⁴⁰, 25, S₃, 2, 32), but
  - the linear programming bound for OOAs shows that MÂ â‰¥ 1 208575 886414 878812 860993 894121 / 96985 358767 > 3⁴⁰ [i]