MinT - Best Known (34, 34+210, s)-Nets in Base 3

Best Known (34, 34+210, s)-Nets in Base 3

(34, 34+210, 38)-Net over F₃ — Constructive and digital

Digital (34, 244, 38)-net over F₃, using

t-expansion [i] based on digital (32, 244, 38)-net over F₃, using
- net from sequence [i] based on digital (32, 37)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 32 and N(F) ≥ 38, using
    - an explicitly constructive algebraic function field [i]

(34, 34+210, 46)-Net over F₃ — Digital

Digital (34, 244, 46)-net over F₃, using

t-expansion [i] based on digital (33, 244, 46)-net over F₃, using
- net from sequence [i] based on digital (33, 45)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 33 and N(F) ≥ 46, using
    - a curve from the manYPoints database [i]

(34, 34+210, 80)-Net in Base 3 — Upper bound on s

There is no (34, 244, 81)-net in base 3, because

6 times m-reduction [i] would yield (34, 238, 81)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3²³⁸, 81, S₃, 3, 204), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 87 189642 485960 958202 911070 585860 771696 964072 404731 750085 525219 437990 967093 723439 943475 549906 831683 116791 055225 665627 / 205 > 3²³⁸ [i]