MinT - Best Known (25, ∞, s)-Nets in Base 3

Best Known (25, ∞, s)-Nets in Base 3

(25, ∞, 32)-Net over F₃ — Constructive and digital

Digital (25, m, 32)-net over F₃ for arbitrarily large m, using

net from sequence [i] based on digital (25, 31)-sequence over F₃, using
- t-expansion [i] based on digital (21, 31)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 21 and N(F) ≥ 32, using
    - an explicitly constructive algebraic function field [i]

(25, ∞, 36)-Net over F₃ — Digital

Digital (25, m, 36)-net over F₃ for arbitrarily large m, using

net from sequence [i] based on digital (25, 35)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 25 and N(F) ≥ 36, using
  - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(25, ∞, 60)-Net in Base 3 — Upper bound on s

There is no (25, m, 61)-net in base 3 for arbitrarily large m, because

m-reduction [i] would yield (25, 239, 61)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3²³⁹, 61, S₃, 4, 214), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 293 861387 637868 414683 885460 122715 934237 915947 734466 268806 770184 031747 333538 104927 216899 075611 914191 245480 963908 724891 / 215 > 3²³⁹ [i]