MinT - Best Known (27, 27+∞, s)-Nets in Base 3

Best Known (27, 27+∞, s)-Nets in Base 3

(27, 27+∞, 37)-Net over F₃ — Constructive and digital

Digital (27, m, 37)-net over F₃ for arbitrarily large m, using

net from sequence [i] based on digital (27, 36)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 26, N(F)Â =Â 36, and 1 place with degree 2 [i] based on function field F/F₃ with g(F) = 26 and N(F) ≥ 36, using an explicitly constructive algebraic function field [i]

(27, 27+∞, 39)-Net over F₃ — Digital

Digital (27, m, 39)-net over F₃ for arbitrarily large m, using

net from sequence [i] based on digital (27, 38)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 27 and N(F) ≥ 39, using
  - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(27, 27+∞, 64)-Net in Base 3 — Upper bound on s

There is no (27, m, 65)-net in base 3 for arbitrarily large m, because

m-reduction [i] would yield (27, 255, 65)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3²⁵⁵, 65, S₃, 4, 228), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 12371 752261 565881 216079 641024 279855 977165 477721 959093 281252 829374 273210 464049 212798 749157 454715 755247 441238 233108 852049 761369 / 229 > 3²⁵⁵ [i]