Information on Result #599876
There is no digital (5, 41, 77)-net over F9, because extracting embedded orthogonal array would yield linear OA(941, 77, F9, 36) (dual of [77, 36, 37]-code), but
- construction Y1 [i] would yield
- linear OA(940, 45, F9, 36) (dual of [45, 5, 37]-code), but
- construction Y1 [i] would yield
- OA(939, 41, S9, 36), but
- the (dual) Plotkin bound shows that M ≥ 739 044147 071729 616580 416051 031916 488005 / 37 > 939 [i]
- OA(95, 45, S9, 4), but
- discarding factors would yield OA(95, 44, S9, 4), but
- the Rao or (dual) Hamming bound shows that M ≥ 60897 > 95 [i]
- discarding factors would yield OA(95, 44, S9, 4), but
- OA(939, 41, S9, 36), but
- construction Y1 [i] would yield
- linear OA(936, 77, F9, 32) (dual of [77, 41, 33]-code), but
- discarding factors / shortening the dual code would yield linear OA(936, 50, F9, 32) (dual of [50, 14, 33]-code), but
- construction Y1 [i] would yield
- linear OA(935, 38, F9, 32) (dual of [38, 3, 33]-code), but
- “Mas†bound on codes from Brouwer’s database [i]
- linear OA(914, 50, F9, 12) (dual of [50, 36, 13]-code), but
- discarding factors / shortening the dual code would yield linear OA(914, 44, F9, 12) (dual of [44, 30, 13]-code), but
- construction Y1 [i] would yield
- linear OA(913, 17, F9, 12) (dual of [17, 4, 13]-code), but
- “MPa†bound on codes from Brouwer’s database [i]
- linear OA(930, 44, F9, 27) (dual of [44, 14, 28]-code), but
- discarding factors / shortening the dual code would yield linear OA(930, 39, F9, 27) (dual of [39, 9, 28]-code), but
- residual code [i] would yield OA(93, 11, S9, 3), but
- discarding factors / shortening the dual code would yield linear OA(930, 39, F9, 27) (dual of [39, 9, 28]-code), but
- linear OA(913, 17, F9, 12) (dual of [17, 4, 13]-code), but
- construction Y1 [i] would yield
- discarding factors / shortening the dual code would yield linear OA(914, 44, F9, 12) (dual of [44, 30, 13]-code), but
- linear OA(935, 38, F9, 32) (dual of [38, 3, 33]-code), but
- construction Y1 [i] would yield
- discarding factors / shortening the dual code would yield linear OA(936, 50, F9, 32) (dual of [50, 14, 33]-code), but
- linear OA(940, 45, F9, 36) (dual of [45, 5, 37]-code), but
Mode: Bound (linear).
Optimality
Show details for fixed k and m, k and s, k and t, m and s, m and t, t and s.
Other Results with Identical Parameters
None.
Depending Results
The following results depend on this result:
| Result | This result only | Method | ||
|---|---|---|---|---|
| 1 | No digital (5, 42, 77)-net over F9 | [i] | m-Reduction | |
| 2 | No digital (5, 43, 77)-net over F9 | [i] | ||
| 3 | No digital (5, 44, 77)-net over F9 | [i] | ||
| 4 | No digital (5, 45, 77)-net over F9 | [i] | ||
| 5 | No digital (5, 46, 77)-net over F9 | [i] | ||