MinT - Best Known (72−52, 72, s)-Nets in Base 3

Best Known (72−52, 72, s)-Nets in Base 3

(72−52, 72, 28)-Net over F₃ — Constructive and digital

Digital (20, 72, 28)-net over F₃, using

t-expansion [i] based on digital (15, 72, 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]

(72−52, 72, 32)-Net over F₃ — Digital

Digital (20, 72, 32)-net over F₃, using

t-expansion [i] based on digital (19, 72, 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]

(72−52, 72, 68)-Net over F₃ — Upper bound on s (digital)

There is no digital (20, 72, 69)-net over F₃, because

10 times m-reduction [i] would yield digital (20, 62, 69)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3⁶², 69, F₃, 42) (dual of [69, 7, 43]-code), but
  - â€œGurâ€ bound on codes from Brouwerâ€™s database [i]

(72−52, 72, 70)-Net in Base 3 — Upper bound on s

There is no (20, 72, 71)-net in base 3, because

9 times m-reduction [i] would yield (20, 63, 71)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3⁶³, 71, S₃, 43), but
  - the linear programming bound shows that MÂ â‰¥ 16965 831756 065304 186694 432350 137421 / 9724 > 3⁶³ [i]