MinT - Best Known (39, 39+172, s)-Nets in Base 3

Best Known (39, 39+172, s)-Nets in Base 3

(39, 39+172, 42)-Net over F₃ — Constructive and digital

Digital (39, 211, 42)-net over F₃, using

net from sequence [i] based on digital (39, 41)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 39 and N(F) ≥ 42, using
  - an explicitly constructive algebraic function field [i]

(39, 39+172, 52)-Net over F₃ — Digital

Digital (39, 211, 52)-net over F₃, using

t-expansion [i] based on digital (37, 211, 52)-net over F₃, using
- net from sequence [i] based on digital (37, 51)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 37 and N(F) ≥ 52, using
    - a curve from the manYPoints database [i]

(39, 39+172, 98)-Net in Base 3 — Upper bound on s

There is no (39, 211, 99)-net in base 3, because

19 times m-reduction [i] would yield (39, 192, 99)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3¹⁹², 99, S₃, 2, 153), but
  - the LP bound with quadratic polynomials shows that MÂ â‰¥ 6922 723989 906201 121312 544744 416752 962627 483676 837442 281916 219853 282245 990568 918349 647886 532011 / 154 > 3¹⁹² [i]