MinT - Best Known (31, 31+58, s)-Nets in Base 3

Best Known (31, 31+58, s)-Nets in Base 3

(31, 31+58, 37)-Net over F₃ — Constructive and digital

Digital (31, 89, 37)-net over F₃, using

t-expansion [i] based on digital (27, 89, 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+58, 42)-Net over F₃ — Digital

Digital (31, 89, 42)-net over F₃, using

t-expansion [i] based on digital (29, 89, 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+58, 104)-Net in Base 3 — Upper bound on s

There is no (31, 89, 105)-net in base 3, because

extracting embedded orthogonal array [i] would yield OA(3⁸⁹, 105, S₃, 58), but
- the linear programming bound shows that MÂ â‰¥ 256 129902 432927 359640 404231 835988 695464 485658 001855 / 65 963711 > 3⁸⁹ [i]