MinT - Best Known (60−40, 60, s)-Nets in Base 3

Best Known (60−40, 60, s)-Nets in Base 3

(60−40, 60, 28)-Net over F₃ — Constructive and digital

Digital (20, 60, 28)-net over F₃, using

t-expansion [i] based on digital (15, 60, 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]

(60−40, 60, 32)-Net over F₃ — Digital

Digital (20, 60, 32)-net over F₃, using

t-expansion [i] based on digital (19, 60, 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]

(60−40, 60, 70)-Net over F₃ — Upper bound on s (digital)

There is no digital (20, 60, 71)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3⁶⁰, 71, F₃, 40) (dual of [71, 11, 41]-code), but
- â€œGurâ€ bound on codes from Brouwerâ€™s database [i]

(60−40, 60, 72)-Net in Base 3 — Upper bound on s

There is no (20, 60, 73)-net in base 3, because

1 times m-reduction [i] would yield (20, 59, 73)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3⁵⁹, 73, S₃, 39), but
  - the linear programming bound shows that MÂ â‰¥ 575447 637759 001874 791324 271023 513509 / 37 398920 > 3⁵⁹ [i]