MinT - Best Known (2, 2+6, s)-Nets in Base 4

Best Known (2, 2+6, s)-Nets in Base 4

(2, 2+6, 12)-Net over F₄ — Constructive and digital

Digital (2, 8, 12)-net over F₄, using

linear code embedding found in a computer search [i]

(2, 2+6, 17)-Net over F₄ — Upper bound on s (digital)

There is no digital (2, 8, 18)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4⁸, 18, F₄, 6) (dual of [18, 10, 7]-code), but
- constructionÂ Y1 [i] would yield
  1. linear OA(4⁷, 10, F₄, 6) (dual of [10, 3, 7]-code), but
    - bound for almost-MDS-codes with dimension nÂ =Â 3 [i]
  2. linear OA(4¹⁰, 18, F₄, 8) (dual of [18, 8, 9]-code), but
    - discarding factors / shortening the dual code would yield linear OA(4¹⁰, 15, F₄, 8) (dual of [15, 5, 9]-code), but
      - residual code [i] would yield OA(4², 6, S₄, 2), but
        bound for OAs with strength kÂ =Â 2 [i]
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 19 > 4² [i]

(2, 2+6, 21)-Net in Base 4 — Upper bound on s

There is no (2, 8, 22)-net in base 4, because

extracting embedded OOA [i] would yield OOA(4⁸, 22, S₄, 2, 6), but
- the linear programming bound for OOAs shows that MÂ â‰¥ 61553 303552 / 938313 > 4⁸ [i]