MinT - Best Known (10, 10+31, s)-Nets in Base 3

Best Known (10, 10+31, s)-Nets in Base 3

(10, 10+31, 19)-Net over F₃ — Constructive and digital

Digital (10, 41, 19)-net over F₃, using

t-expansion [i] based on digital (9, 41, 19)-net over F₃, using
- net from sequence [i] based on digital (9, 18)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 9 and N(F) ≥ 19, using
    - an explicitly constructive algebraic function field [i]

(10, 10+31, 20)-Net over F₃ — Digital

Digital (10, 41, 20)-net over F₃, using

net from sequence [i] based on digital (10, 19)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 10 and N(F) ≥ 20, using
  - a curve from the manYPoints database [i]

(10, 10+31, 36)-Net over F₃ — Upper bound on s (digital)

There is no digital (10, 41, 37)-net over F₃, because

10 times m-reduction [i] would yield digital (10, 31, 37)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3³¹, 37, F₃, 21) (dual of [37, 6, 22]-code), but
  - â€œBouâ€ bound on codes from Brouwerâ€™s database [i]

(10, 10+31, 39)-Net in Base 3 — Upper bound on s

There is no (10, 41, 40)-net in base 3, because

7 times m-reduction [i] would yield (10, 34, 40)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3³⁴, 40, S₃, 24), but
  - the linear programming bound shows that MÂ â‰¥ 2 701703 435345 984178 / 155 > 3³⁴ [i]