MinT - Best Known (85−58, 85, s)-Nets in Base 3

Best Known (85−58, 85, s)-Nets in Base 3

(85−58, 85, 37)-Net over F₃ — Constructive and digital

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]

(85−58, 85, 39)-Net over F₃ — Digital

Digital (27, 85, 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]

(85−58, 85, 89)-Net over F₃ — Upper bound on s (digital)

There is no digital (27, 85, 90)-net over F₃, because

4 times m-reduction [i] would yield digital (27, 81, 90)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3⁸¹, 90, F₃, 54) (dual of [90, 9, 55]-code), but
  - residual code [i] would yield linear OA(3²⁷, 35, F₃, 18) (dual of [35, 8, 19]-code), but
    - residual code [i] would yield linear OA(3⁹, 16, F₃, 6) (dual of [16, 7, 7]-code), but
      - â€œvE2â€ bound on codes from Brouwerâ€™s database [i]

(85−58, 85, 90)-Net in Base 3 — Upper bound on s

There is no (27, 85, 91)-net in base 3, because

2 times m-reduction [i] would yield (27, 83, 91)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3⁸³, 91, S₃, 56), but
  - the linear programming bound shows that MÂ â‰¥ 969773 729787 523602 876821 942164 080815 560161 / 209 > 3⁸³ [i]