MinT - Best Known (209−165, 209, s)-Nets in Base 3

Best Known (209−165, 209, s)-Nets in Base 3

(209−165, 209, 42)-Net over F₃ — Constructive and digital

Digital (44, 209, 42)-net over F₃, using

t-expansion [i] based on digital (39, 209, 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]

(209−165, 209, 56)-Net over F₃ — Digital

Digital (44, 209, 56)-net over F₃, using

t-expansion [i] based on digital (40, 209, 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]

(209−165, 209, 140)-Net in Base 3 — Upper bound on s

There is no (44, 209, 141)-net in base 3, because

81 times m-reduction [i] would yield (44, 128, 141)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3¹²⁸, 141, S₃, 84), but
  - the linear programming bound shows that MÂ â‰¥ 8 586449 512614 255704 906677 599653 756381 172136 834518 319080 363152 884431 / 524875 > 3¹²⁸ [i]