MinT - Best Known (19, 55, s)-Nets in Base 3

Best Known (19, 55, s)-Nets in Base 3

(19, 55, 28)-Net over F₃ — Constructive and digital

Digital (19, 55, 28)-net over F₃, using

t-expansion [i] based on digital (15, 55, 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]

(19, 55, 32)-Net over F₃ — Digital

Digital (19, 55, 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]

(19, 55, 69)-Net over F₃ — Upper bound on s (digital)

There is no digital (19, 55, 70)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3⁵⁵, 70, F₃, 36) (dual of [70, 15, 37]-code), but
- constructionÂ Y1 [i] would yield
  1. linear OA(3⁵⁴, 62, F₃, 36) (dual of [62, 8, 37]-code), but
    - constructionÂ Y1 [i] would yield
      1. linear OA(3⁵³, 58, F₃, 36) (dual of [58, 5, 37]-code), but
        residual code [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, 62, 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]
  2. OA(3¹⁵, 70, S₃, 8), but
    - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 15 118041 > 3¹⁵ [i]

(19, 55, 72)-Net in Base 3 — Upper bound on s

There is no (19, 55, 73)-net in base 3, because

extracting embedded orthogonal array [i] would yield OA(3⁵⁵, 73, S₃, 36), but
- the linear programming bound shows that MÂ â‰¥ 418845 330644 374396 228543 975709 282088 / 1987 676075 > 3⁵⁵ [i]