MinT - Best Known (4, 10, s)-Nets in Base 3

Best Known (4, 10, s)-Nets in Base 3

(4, 10, 19)-Net over F₃ — Constructive and digital

Digital (4, 10, 19)-net over F₃, using

linear code embedding found in a computer search [i]

(4, 10, 27)-Net over F₃ — Upper bound on s (digital)

There is no digital (4, 10, 28)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹⁰, 28, F₃, 6) (dual of [28, 18, 7]-code), but
- constructionÂ Y1 [i] would yield
  1. linear OA(3⁹, 16, F₃, 6) (dual of [16, 7, 7]-code), but
    - â€œvE2â€ bound on codes from Brouwerâ€™s database [i]
  2. linear OA(3¹⁸, 28, F₃, 12) (dual of [28, 10, 13]-code), but
    - residual code [i] would yield linear OA(3⁶, 15, F₃, 4) (dual of [15, 9, 5]-code), but
      - â€œvE2â€ bound on codes from Brouwerâ€™s database [i]

(4, 10, 32)-Net in Base 3 — Upper bound on s

There is no (4, 10, 33)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 59291 > 3¹⁰ [i]