MinT - Best Known (68−59, 68, s)-Nets in Base 3

Best Known (68−59, 68, s)-Nets in Base 3

(68−59, 68, 19)-Net over F₃ — Constructive and digital

Digital (9, 68, 19)-net over F₃, using

net from sequence [i] based on digital (9, 18)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 9 and N(F) ≥ 19, using
  - an explicitly constructive algebraic function field [i]

(68−59, 68, 28)-Net in Base 3 — Upper bound on s

There is no (9, 68, 29)-net in base 3, because

15 times m-reduction [i] would yield (9, 53, 29)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3⁵³, 29, S₃, 2, 44), but
  - the LP bound with quadratic polynomials shows that MÂ â‰¥ 109 838392 116853 446081 848097 / 5 > 3⁵³ [i]