MinT - Best Known (2, 7, s)-Nets in Base 5

Best Known (2, 7, s)-Nets in Base 5

(2, 7, 21)-Net over F₅ — Constructive and digital

Digital (2, 7, 21)-net over F₅, using

linear code embedding found in a computer search [i]

(2, 7, 41)-Net over F₅ — Upper bound on s (digital)

There is no digital (2, 7, 42)-net over F₅, because

extracting embedded orthogonal array [i] would yield linear OA(5⁷, 42, F₅, 5) (dual of [42, 35, 6]-code), but
- constructionÂ Y1 [i] would yield
  1. linear OA(5⁶, 13, F₅, 5) (dual of [13, 7, 6]-code), but
    - dual of a near-MDS code is again a near-MDS code [i] would yield linear OA(5⁷, 13, F₅, 6) (dual of [13, 6, 7]-code), but
      - discarding factors / shortening the dual code would yield linear OA(5⁷, 12, F₅, 6) (dual of [12, 5, 7]-code), but
        a result by Dodunekov and Landjev [i]
  2. linear OA(5³⁵, 42, F₅, 29) (dual of [42, 7, 30]-code), but
    - discarding factors / shortening the dual code would yield linear OA(5³⁵, 41, F₅, 29) (dual of [41, 6, 30]-code), but
      - â€œDG1â€ bound on codes from Brouwerâ€™s database [i]

(2, 7, 43)-Net in Base 5 — Upper bound on s

There is no (2, 7, 44)-net in base 5, because

1 times m-reduction [i] would yield (2, 6, 44)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 16193 > 5⁶ [i]
- extracting embedded orthogonal array [i] would yield OA(5⁶, 44, S₅, 4), but
  - the linear programming bound shows that MÂ â‰¥ 439375 / 27 > 5⁶ [i]