MinT - Best Known (165−138, 165, s)-Nets in Base 3

Best Known (165−138, 165, s)-Nets in Base 3

(165−138, 165, 37)-Net over F₃ — Constructive and digital

Digital (27, 165, 37)-net over F₃, 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]

(165−138, 165, 39)-Net over F₃ — Digital

Digital (27, 165, 39)-net over F₃, 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]

(165−138, 165, 70)-Net in Base 3 — Upper bound on s

There is no (27, 165, 71)-net in base 3, because

28 times m-reduction [i] would yield (27, 137, 71)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3¹³⁷, 71, S₃, 2, 110), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 10 442979 136963 283965 427040 323903 217220 344490 744684 442124 765996 751335 / 37 > 3¹³⁷ [i]