MinT - Best Known (222−186, 222, s)-Nets in Base 3

Best Known (222−186, 222, s)-Nets in Base 3

(222−186, 222, 38)-Net over F₃ — Constructive and digital

Digital (36, 222, 38)-net over F₃, using

t-expansion [i] based on digital (32, 222, 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]

(222−186, 222, 48)-Net over F₃ — Digital

Digital (36, 222, 48)-net over F₃, using

net from sequence [i] based on digital (36, 47)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 36 and N(F) ≥ 48, using
  - a curve from the manYPoints database [i]

(222−186, 222, 91)-Net in Base 3 — Upper bound on s

There is no (36, 222, 92)-net in base 3, because

44 times m-reduction [i] would yield (36, 178, 92)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3¹⁷⁸, 92, S₃, 2, 142), but
  - the LP bound with quadratic polynomials shows that MÂ â‰¥ 1295 014916 779728 669767 925371 250464 549757 221623 840718 344308 026617 664271 334831 561646 333217 / 143 > 3¹⁷⁸ [i]