MinT - Best Known (4, 4+9, s)-Nets in Base 3

Best Known (4, 4+9, s)-Nets in Base 3

(4, 4+9, 13)-Net over F₃ — Constructive and digital

Digital (4, 13, 13)-net over F₃, using

a shift-net [i]

(4, 4+9, 18)-Net over F₃ — Upper bound on s (digital)

There is no digital (4, 13, 19)-net over F₃, because

extracting embedded orthogonal array [i] would yield 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]

(4, 4+9, 25)-Net in Base 3 — Upper bound on s

There is no (4, 13, 26)-net in base 3, because

extracting embedded OOA [i] would yield OOA(3¹³, 26, S₃, 2, 9), but
- the linear programming bound for OOAs shows that MÂ â‰¥ 163 695257 861120 211057 749767 575918 117639 / 100 569955 229419 122359 165135 924498 > 3¹³ [i]