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

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

(4, 68, 15)-Net over F₄ — Constructive and digital

Digital (4, 68, 15)-net over F₄, using

net from sequence [i] based on digital (4, 14)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 4 and N(F) ≥ 15, using
  - an explicitly constructive algebraic function field [i]

(4, 68, 20)-Net in Base 4 — Upper bound on s

There is no (4, 68, 21)-net in base 4, because

44 times m-reduction [i] would yield (4, 24, 21)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4²⁴, 21, S₄, 2, 20), but
  - the linear programming bound for OOAs shows that MÂ â‰¥ 106960 491150 049280 / 311 > 4²⁴ [i]