MinT - Best Known (137−117, 137, s)-Nets in Base 3

Best Known (137−117, 137, s)-Nets in Base 3

(137−117, 137, 28)-Net over F₃ — Constructive and digital

Digital (20, 137, 28)-net over F₃, using

t-expansion [i] based on digital (15, 137, 28)-net over F₃, using
- net from sequence [i] based on digital (15, 27)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, using
    - an explicitly constructive algebraic function field [i]

(137−117, 137, 32)-Net over F₃ — Digital

Digital (20, 137, 32)-net over F₃, using

t-expansion [i] based on digital (19, 137, 32)-net over F₃, using
- net from sequence [i] based on digital (19, 31)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 19 and N(F) ≥ 32, using
    - a curve from the manYPoints database [i]

(137−117, 137, 54)-Net in Base 3 — Upper bound on s

There is no (20, 137, 55)-net in base 3, because

32 times m-reduction [i] would yield (20, 105, 55)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3¹⁰⁵, 55, S₃, 2, 85), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 6762 783827 045452 685860 442944 784482 340397 054903 697122 / 43 > 3¹⁰⁵ [i]