MinT - Best Known (31, 31+41, s)-Nets in Base 3

Best Known (31, 31+41, s)-Nets in Base 3

(31, 31+41, 37)-Net over F₃ — Constructive and digital

Digital (31, 72, 37)-net over F₃, using

t-expansion [i] based on digital (27, 72, 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]

(31, 31+41, 42)-Net over F₃ — Digital

Digital (31, 72, 42)-net over F₃, using

t-expansion [i] based on digital (29, 72, 42)-net over F₃, using
- net from sequence [i] based on digital (29, 41)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 29 and N(F) ≥ 42, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(31, 31+41, 185)-Net in Base 3 — Upper bound on s

There is no (31, 72, 186)-net in base 3, because

extracting embedded orthogonal array [i] would yield OA(3⁷², 186, S₃, 41), but
- the linear programming bound shows that MÂ â‰¥ 105100 370922 616110 968895 584169 245447 272448 588453 048973 847719 242504 285147 362460 101173 735850 155186 787120 222934 850113 841926 239160 654704 889261 785133 583696 / 4 604865 149369 910190 713405 030303 912393 944804 135160 770710 332878 788739 361629 708558 657270 076629 943411 022298 655059 309017 > 3⁷² [i]