MinT - Best Known (229−186, 229, s)-Nets in Base 3

Best Known (229−186, 229, s)-Nets in Base 3

(229−186, 229, 42)-Net over F₃ — Constructive and digital

Digital (43, 229, 42)-net over F₃, using

t-expansion [i] based on digital (39, 229, 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]

(229−186, 229, 56)-Net over F₃ — Digital

Digital (43, 229, 56)-net over F₃, using

t-expansion [i] based on digital (40, 229, 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]

(229−186, 229, 107)-Net in Base 3 — Upper bound on s

There is no (43, 229, 108)-net in base 3, because

19 times m-reduction [i] would yield (43, 210, 108)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3²¹⁰, 108, S₃, 2, 167), but
  - the LP bound with quadratic polynomials shows that MÂ â‰¥ 894001 704460 497167 517084 802238 772817 524446 150306 626620 789752 584196 705332 228166 578407 219059 349405 257793 / 56 > 3²¹⁰ [i]