MinT - Best Known (159−116, 159, s)-Nets in Base 3

Best Known (159−116, 159, s)-Nets in Base 3

(159−116, 159, 42)-Net over F₃ — Constructive and digital

Digital (43, 159, 42)-net over F₃, using

t-expansion [i] based on digital (39, 159, 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]

(159−116, 159, 56)-Net over F₃ — Digital

Digital (43, 159, 56)-net over F₃, using

t-expansion [i] based on digital (40, 159, 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]

(159−116, 159, 137)-Net in Base 3 — Upper bound on s

There is no (43, 159, 138)-net in base 3, because

34 times m-reduction [i] would yield (43, 125, 138)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3¹²⁵, 138, S₃, 82), but
  - the linear programming bound shows that MÂ â‰¥ 2 604298 500822 942421 007731 043738 827010 505663 123983 033715 065100 424407 / 4 505323 > 3¹²⁵ [i]