MinT - Best Known (5, 5+9, s)-Nets in Base 3

Best Known (5, 5+9, s)-Nets in Base 3

(5, 5+9, 15)-Net over F₃ — Constructive and digital

Digital (5, 14, 15)-net over F₃, using

linear code embedding found in a computer search [i]

(5, 5+9, 29)-Net over F₃ — Upper bound on s (digital)

There is no digital (5, 14, 30)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹⁴, 30, F₃, 9) (dual of [30, 16, 10]-code), but
- constructionÂ Y1 [i] would yield
  1. linear OA(3¹³, 19, F₃, 9) (dual of [19, 6, 10]-code), but
    - constructionÂ Y1 [i] would yield
      1. linear OA(3¹², 15, F₃, 9) (dual of [15, 3, 10]-code), but
        Griesmer bound [i]
      2. OA(3⁶, 19, S₃, 4), but
        the linear programming bound shows that MÂ â‰¥ 51975 / 67 > 3⁶ [i]
  2. linear OA(3¹⁶, 30, F₃, 11) (dual of [30, 14, 12]-code), but
    - discarding factors / shortening the dual code would yield linear OA(3¹⁶, 24, F₃, 11) (dual of [24, 8, 12]-code), but
      - â€œBSâ€ bound on codes from Brouwerâ€™s database [i]

(5, 5+9, 34)-Net in Base 3 — Upper bound on s

There is no (5, 14, 35)-net in base 3, because

extracting embedded OOA [i] would yield OOA(3¹⁴, 35, S₃, 2, 9), but
- the linear programming bound for OOAs shows that MÂ â‰¥ 85 860836 943400 308679 107400 550256 119100 / 17 045811 826801 425760 656153 573217 > 3¹⁴ [i]