MinT - Best Known (190−164, 190, s)-Nets in Base 3

Best Known (190−164, 190, s)-Nets in Base 3

(190−164, 190, 36)-Net over F₃ — Constructive and digital

Digital (26, 190, 36)-net over F₃, using

net from sequence [i] based on digital (26, 35)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 26 and N(F) ≥ 36, using
  - an explicitly constructive algebraic function field [i]

(190−164, 190, 37)-Net over F₃ — Digital

Digital (26, 190, 37)-net over F₃, using

net from sequence [i] based on digital (26, 36)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 25, N(F)Â =Â 36, and 1 place with degree 2 [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]

(190−164, 190, 63)-Net in Base 3 — Upper bound on s

There is no (26, 190, 64)-net in base 3, because

3 times m-reduction [i] would yield (26, 187, 64)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3¹⁸⁷, 64, S₃, 3, 161), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 203622 051326 600866 722921 442197 205396 558885 494148 602360 843256 185236 270428 117927 310626 393473 > 3¹⁸⁷ [i]