MinT - Best Known (19, 57, s)-Nets in Base 3

Best Known (19, 57, s)-Nets in Base 3

(19, 57, 28)-Net over F₃ — Constructive and digital

Digital (19, 57, 28)-net over F₃, using

t-expansion [i] based on digital (15, 57, 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]

(19, 57, 32)-Net over F₃ — Digital

Digital (19, 57, 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]

(19, 57, 67)-Net over F₃ — Upper bound on s (digital)

There is no digital (19, 57, 68)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3⁵⁷, 68, F₃, 38) (dual of [68, 11, 39]-code), but
- â€œGurâ€ bound on codes from Brouwerâ€™s database [i]

(19, 57, 69)-Net in Base 3 — Upper bound on s

There is no (19, 57, 70)-net in base 3, because

1 times m-reduction [i] would yield (19, 56, 70)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3⁵⁶, 70, S₃, 37), but
  - the linear programming bound shows that MÂ â‰¥ 185 546099 770621 322782 066734 120777 / 337535 > 3⁵⁶ [i]