MinT - Best Known (4, 11, s)-Nets in Base 3

Best Known (4, 11, s)-Nets in Base 3

(4, 11, 15)-Net over F₃ — Constructive and digital

Digital (4, 11, 15)-net over F₃, using

linear code embedding found in a computer search [i]

(4, 11, 26)-Net over F₃ — Upper bound on s (digital)

There is no digital (4, 11, 27)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹¹, 27, F₃, 7) (dual of [27, 16, 8]-code), but
- constructionÂ Y1 [i] would yield
  1. linear OA(3¹⁰, 16, F₃, 7) (dual of [16, 6, 8]-code), but
    - â€œvE2â€ bound on codes from Brouwerâ€™s database [i]
  2. linear OA(3¹⁶, 27, F₃, 11) (dual of [27, 11, 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]

(4, 11, 31)-Net in Base 3 — Upper bound on s

There is no (4, 11, 32)-net in base 3, because

extracting embedded OOA [i] would yield OOA(3¹¹, 32, S₃, 2, 7), but
- the linear programming bound for OOAs shows that MÂ â‰¥ 3 145072 951240 095330 / 16 625272 919921 > 3¹¹ [i]