MinT - Best Known (78−59, 78, s)-Nets in Base 3

Best Known (78−59, 78, s)-Nets in Base 3

(78−59, 78, 28)-Net over F₃ — Constructive and digital

Digital (19, 78, 28)-net over F₃, using

t-expansion [i] based on digital (15, 78, 28)-net over F₃, using
- net from sequence [i] based on digital (15, 27)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, using
    - an explicitly constructive algebraic function field [i]

(78−59, 78, 32)-Net over F₃ — Digital

Digital (19, 78, 32)-net over F₃, using

net from sequence [i] based on digital (19, 31)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 19 and N(F) ≥ 32, using
  - a curve from the manYPoints database [i]

(78−59, 78, 65)-Net over F₃ — Upper bound on s (digital)

There is no digital (19, 78, 66)-net over F₃, because

20 times m-reduction [i] would yield digital (19, 58, 66)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3⁵⁸, 66, F₃, 39) (dual of [66, 8, 40]-code), but
  - constructionÂ Y1 [i] would yield
    1. 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(3⁸, 66, S₃, 4), but
      - discarding factors would yield OA(3⁸, 58, S₃, 4), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 6729 > 3⁸ [i]

(78−59, 78, 66)-Net in Base 3 — Upper bound on s

There is no (19, 78, 67)-net in base 3, because

18 times m-reduction [i] would yield (19, 60, 67)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3⁶⁰, 67, S₃, 41), but
  - the linear programming bound shows that MÂ â‰¥ 367 404168 771298 835858 389852 553067 / 8575 > 3⁶⁰ [i]