MinT - Best Known (166−133, 166, s)-Nets in Base 3

Best Known (166−133, 166, s)-Nets in Base 3

(166−133, 166, 38)-Net over F₃ — Constructive and digital

Digital (33, 166, 38)-net over F₃, using

t-expansion [i] based on digital (32, 166, 38)-net over F₃, using
- net from sequence [i] based on digital (32, 37)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 32 and N(F) ≥ 38, using
    - an explicitly constructive algebraic function field [i]

(166−133, 166, 46)-Net over F₃ — Digital

Digital (33, 166, 46)-net over F₃, using

net from sequence [i] based on digital (33, 45)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 33 and N(F) ≥ 46, using
  - a curve from the manYPoints database [i]

(166−133, 166, 84)-Net in Base 3 — Upper bound on s

There is no (33, 166, 85)-net in base 3, because

2 times m-reduction [i] would yield (33, 164, 85)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3¹⁶⁴, 85, S₃, 2, 131), but
  - the LP bound with quadratic polynomials shows that MÂ â‰¥ 79 633955 442598 930015 320742 953399 407094 656897 632406 980605 562743 292876 388434 707645 / 44 > 3¹⁶⁴ [i]