MinT - Best Known (226−191, 226, s)-Nets in Base 3

Best Known (226−191, 226, s)-Nets in Base 3

(226−191, 226, 38)-Net over F₃ — Constructive and digital

Digital (35, 226, 38)-net over F₃, using

t-expansion [i] based on digital (32, 226, 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]

(226−191, 226, 47)-Net over F₃ — Digital

Digital (35, 226, 47)-net over F₃, using

net from sequence [i] based on digital (35, 46)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 35 and N(F) ≥ 47, using
  - a curve from the manYPoints database [i]

(226−191, 226, 89)-Net in Base 3 — Upper bound on s

There is no (35, 226, 90)-net in base 3, because

52 times m-reduction [i] would yield (35, 174, 90)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3¹⁷⁴, 90, S₃, 2, 139), but
  - the LP bound with quadratic polynomials shows that MÂ â‰¥ 17 868760 652734 092030 203763 292490 070040 223101 563524 799231 555922 829064 019870 587996 572499 / 140 > 3¹⁷⁴ [i]