MinT - Best Known (10, 10+57, s)-Nets in Base 3

Best Known (10, 10+57, s)-Nets in Base 3

(10, 10+57, 19)-Net over F₃ — Constructive and digital

Digital (10, 67, 19)-net over F₃, using

t-expansion [i] based on digital (9, 67, 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+57, 20)-Net over F₃ — Digital

Digital (10, 67, 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+57, 30)-Net in Base 3 — Upper bound on s

There is no (10, 67, 31)-net in base 3, because

9 times m-reduction [i] would yield (10, 58, 31)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3⁵⁸, 31, S₃, 2, 48), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 42391 158275 216203 514294 433201 / 7 > 3⁵⁸ [i]