MinT - Best Known (85−57, 85, s)-Nets in Base 3

Best Known (85−57, 85, s)-Nets in Base 3

Digital (28, 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]

Digital (28, 85, 39)-net over F₃, using

There is no digital (28, 85, 94)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3⁸⁵, 94, F₃, 57) (dual of [94, 9, 58]-code), but
- residual code [i] would yield linear OA(3²⁸, 36, F₃, 19) (dual of [36, 8, 20]-code), but
  - 1 times truncation [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]

There is no (28, 85, 95)-net in base 3, because

2 times m-reduction [i] would yield (28, 83, 95)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3⁸³, 95, S₃, 55), but
  - the linear programming bound shows that MÂ â‰¥ 648 778625 227853 290324 593879 307770 065609 747709 / 118144 > 3⁸³ [i]