MinT - Best Known (85−55, 85, s)-Nets in Base 3

Best Known (85−55, 85, s)-Nets in Base 3

(85−55, 85, 37)-Net over F₃ — Constructive and digital

Digital (30, 85, 37)-net over F₃, using

t-expansion [i] based on digital (27, 85, 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]

(85−55, 85, 42)-Net over F₃ — Digital

Digital (30, 85, 42)-net over F₃, using

t-expansion [i] based on digital (29, 85, 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]

(85−55, 85, 103)-Net in Base 3 — Upper bound on s

There is no (30, 85, 104)-net in base 3, because

extracting embedded orthogonal array [i] would yield OA(3⁸⁵, 104, S₃, 55), but
- the linear programming bound shows that MÂ â‰¥ 4744 075529 079928 845960 058598 807593 969359 823300 884489 / 94629 133984 > 3⁸⁵ [i]