MinT - Best Known (227−186, 227, s)-Nets in Base 3

Best Known (227−186, 227, s)-Nets in Base 3

(227−186, 227, 42)-Net over F₃ — Constructive and digital

Digital (41, 227, 42)-net over F₃, using

t-expansion [i] based on digital (39, 227, 42)-net over F₃, using
- net from sequence [i] based on digital (39, 41)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 39 and N(F) ≥ 42, using
    - an explicitly constructive algebraic function field [i]

(227−186, 227, 56)-Net over F₃ — Digital

Digital (41, 227, 56)-net over F₃, using

t-expansion [i] based on digital (40, 227, 56)-net over F₃, using
- net from sequence [i] based on digital (40, 55)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 40 and N(F) ≥ 56, using
    - a curve from the manYPoints database [i]

(227−186, 227, 102)-Net in Base 3 — Upper bound on s

There is no (41, 227, 103)-net in base 3, because

26 times m-reduction [i] would yield (41, 201, 103)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3²⁰¹, 103, S₃, 2, 160), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 21 514733 098945 856316 441287 084898 149773 167909 664924 954173 940571 900866 491763 005476 822545 160622 564081 / 23 > 3²⁰¹ [i]