MinT - Best Known (12, 12+209, s)-Nets in Base 4

Best Known (12, 12+209, s)-Nets in Base 4

(12, 12+209, 28)-Net over F₄ — Constructive and digital

Digital (12, 221, 28)-net over F₄, using

net from sequence [i] based on digital (12, 27)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 12 and N(F) ≥ 28, using
  - an explicitly constructive algebraic function field [i]

(12, 12+209, 29)-Net over F₄ — Digital

Digital (12, 221, 29)-net over F₄, using

net from sequence [i] based on digital (12, 28)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 12 and N(F) ≥ 29, using
  - a curve from the manYPoints database [i]

(12, 12+209, 47)-Net in Base 4 — Upper bound on s

There is no (12, 221, 48)-net in base 4, because

81 times m-reduction [i] would yield (12, 140, 48)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4¹⁴⁰, 48, S₄, 3, 128), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 124 330809 102446 660538 845562 036705 210025 114037 699336 929360 115994 223289 874253 133343 883264 / 43 > 4¹⁴⁰ [i]