MinT - Best Known (2, 68, s)-Nets in Base 4

Best Known (2, 68, s)-Nets in Base 4

(2, 68, 10)-Net over F₄ — Constructive and digital

Digital (2, 68, 10)-net over F₄, using

net from sequence [i] based on digital (2, 9)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 2 and N(F) ≥ 10, using
  - an explicitly constructive algebraic function field [i]

(2, 68, 12)-Net in Base 4 — Upper bound on s

There is no (2, 68, 13)-net in base 4, because

52 times m-reduction [i] would yield (2, 16, 13)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4¹⁶, 13, S₄, 3, 14), but
  - the linear programming bound for OOAs shows that MÂ â‰¥ 176093 659136 / 37 > 4¹⁶ [i]