MinT - Best Known (216−157, 216, s)-Nets in Base 3

Best Known (216−157, 216, s)-Nets in Base 3

(216−157, 216, 48)-Net over F₃ — Constructive and digital

Digital (59, 216, 48)-net over F₃, using

t-expansion [i] based on digital (45, 216, 48)-net over F₃, using
- net from sequence [i] based on digital (45, 47)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
    - an explicitly constructive algebraic function field [i]

(216−157, 216, 64)-Net over F₃ — Digital

Digital (59, 216, 64)-net over F₃, using

t-expansion [i] based on digital (49, 216, 64)-net over F₃, using
- net from sequence [i] based on digital (49, 63)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 49 and N(F) ≥ 64, using
    - a curve from the manYPoints database [i]

(216−157, 216, 187)-Net over F₃ — Upper bound on s (digital)

There is no digital (59, 216, 188)-net over F₃, because

37 times m-reduction [i] would yield digital (59, 179, 188)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁷⁹, 188, F₃, 120) (dual of [188, 9, 121]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(3¹⁷⁸, 184, F₃, 120) (dual of [184, 6, 121]-code), but
      - residual code [i] would yield linear OA(3⁵⁸, 63, F₃, 40) (dual of [63, 5, 41]-code), but
        1 times truncation [i] would yield linear OA(3⁵⁷, 62, F₃, 39) (dual of [62, 5, 40]-code), but
        residual code [i] would yield linear OA(3¹⁸, 22, F₃, 13) (dual of [22, 4, 14]-code), but
        1 times truncation [i] would yield linear OA(3¹⁷, 21, F₃, 12) (dual of [21, 4, 13]-code), but
        â€œHNâ€ bound on codes from Brouwerâ€™s database [i]
    2. OA(3⁹, 188, S₃, 4), but
      - discarding factors would yield OA(3⁹, 100, S₃, 4), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 20001 > 3⁹ [i]

(216−157, 216, 190)-Net in Base 3 — Upper bound on s

There is no (59, 216, 191)-net in base 3, because

30 times m-reduction [i] would yield (59, 186, 191)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3¹⁸⁶, 191, S₃, 127), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 4 498196 224760 364601 242719 132174 628305 800834 098010 033971 355568 455673 974002 968757 862019 419449 / 64 > 3¹⁸⁶ [i]