MinT - Best Known (68−64, 68, s)-Nets in Base 3

Best Known (68−64, 68, s)-Nets in Base 3

(68−64, 68, 12)-Net over F₃ — Constructive and digital

Digital (4, 68, 12)-net over F₃, using

net from sequence [i] based on digital (4, 11)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 4 and N(F) ≥ 12, using
  - an explicitly constructive algebraic function field [i]

(68−64, 68, 14)-Net in Base 3 — Upper bound on s

There is no (4, 68, 15)-net in base 3, because

46 times m-reduction [i] would yield (4, 22, 15)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3²², 15, S₃, 3, 18), but
  - the linear programming bound for OOAs shows that MÂ â‰¥ 84237 806413 134527 140642 081122 / 2 548701 622307 460229 > 3²² [i]