MinT - Best Known (57−37, 57, s)-Nets in Base 3

Best Known (57−37, 57, s)-Nets in Base 3

(57−37, 57, 28)-Net over F₃ — Constructive and digital

Digital (20, 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]

(57−37, 57, 32)-Net over F₃ — Digital

Digital (20, 57, 32)-net over F₃, using

t-expansion [i] based on 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]

(57−37, 57, 77)-Net in Base 3 — Upper bound on s

There is no (20, 57, 78)-net in base 3, because

extracting embedded orthogonal array [i] would yield OA(3⁵⁷, 78, S₃, 37), but
- the linear programming bound shows that MÂ â‰¥ 4 189427 328087 433604 993587 700284 480257 / 2119 390625 > 3⁵⁷ [i]