MinT - Information on Result #599907

Information on Result #599907

There is no digital (4, 68, 77)-net over F₁₆, because extracting embedded orthogonal array would yield linear OA(16⁶⁸, 77, F₁₆, 64) (dual of [77, 9, 65]-code), but

constructionÂ Y1 [i] would yield
1. OA(16⁶⁷, 69, S₁₆, 64), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 7588 550360 256754 183279 148073 529370 729071 901715 047420 004889 892225 542594 864082 845696 / 13 > 16⁶⁷ [i]
2. OA(16⁹, 77, S₁₆, 8), but
  - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 68757 087631 > 16⁹ [i]

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		Method
1	No digital (4, 69, 77)-net over F₁₆	[i]	m-Reduction
2	No digital (4, 70, 77)-net over F₁₆	[i]
3	No digital (4, 71, 77)-net over F₁₆	[i]
4	No digital (4, 72, 77)-net over F₁₆	[i]
5	No digital (4, 73, 77)-net over F₁₆	[i]
6	No digital (4, 74, 77)-net over F₁₆	[i]
7	No digital (4, 75, 77)-net over F₁₆	[i]
8	No digital (4, 76, 77)-net over F₁₆	[i]
9	No digital (4, 77, 77)-net over F₁₆	[i]
10	No digital (4, 78, 77)-net over F₁₆	[i]
11	No digital (4, 79, 77)-net over F₁₆	[i]
12	No digital (4, 80, 77)-net over F₁₆	[i]
13	No digital (4, 81, 77)-net over F₁₆	[i]
14	No digital (4, 82, 77)-net over F₁₆	[i]
15	No digital (4, 83, 77)-net over F₁₆	[i]
16	No digital (4, 84, 77)-net over F₁₆	[i]
17	No digital (4, 85, 77)-net over F₁₆	[i]
18	No digital (4, 86, 77)-net over F₁₆	[i]
19	No digital (4, 87, 77)-net over F₁₆	[i]
20	No digital (4, 88, 77)-net over F₁₆	[i]
21	No digital (4, 89, 77)-net over F₁₆	[i]
22	No digital (4, 90, 77)-net over F₁₆	[i]
23	No digital (4, 91, 77)-net over F₁₆	[i]
24	No digital (4, 92, 77)-net over F₁₆	[i]
25	No digital (4, 93, 77)-net over F₁₆	[i]
26	No digital (4, 94, 77)-net over F₁₆	[i]
27	No digital (4, 95, 77)-net over F₁₆	[i]
28	No digital (4, 96, 77)-net over F₁₆	[i]
29	No digital (4, 97, 77)-net over F₁₆	[i]
30	No digital (4, 98, 77)-net over F₁₆	[i]
31	No digital (4, 99, 77)-net over F₁₆	[i]
32	No digital (4, 100, 77)-net over F₁₆	[i]
33	No digital (4, 101, 77)-net over F₁₆	[i]
34	No digital (4, 102, 77)-net over F₁₆	[i]
35	No digital (4, 103, 77)-net over F₁₆	[i]
36	No digital (4, 104, 77)-net over F₁₆	[i]
37	No digital (4, 105, 77)-net over F₁₆	[i]
38	No digital (4, 106, 77)-net over F₁₆	[i]
39	No digital (4, 107, 77)-net over F₁₆	[i]
40	No digital (4, 108, 77)-net over F₁₆	[i]
41	No digital (4, 109, 77)-net over F₁₆	[i]
42	No digital (4, 110, 77)-net over F₁₆	[i]
43	No digital (4, 111, 77)-net over F₁₆	[i]
44	No digital (4, 112, 77)-net over F₁₆	[i]
45	No digital (4, 113, 77)-net over F₁₆	[i]
46	No digital (4, 114, 77)-net over F₁₆	[i]
47	No digital (4, 115, 77)-net over F₁₆	[i]
48	No digital (4, 116, 77)-net over F₁₆	[i]
49	No digital (4, 117, 77)-net over F₁₆	[i]
50	No digital (4, 118, 77)-net over F₁₆	[i]
51	No digital (4, 119, 77)-net over F₁₆	[i]
52	No digital (4, 120, 77)-net over F₁₆	[i]
53	No digital (4, 121, 77)-net over F₁₆	[i]
54	No digital (4, 122, 77)-net over F₁₆	[i]
55	No digital (4, 123, 77)-net over F₁₆	[i]
56	No digital (4, 124, 77)-net over F₁₆	[i]
57	No digital (4, 125, 77)-net over F₁₆	[i]
58	No digital (4, 126, 77)-net over F₁₆	[i]
59	No digital (4, 127, 77)-net over F₁₆	[i]
60	No digital (4, 128, 77)-net over F₁₆	[i]
61	No digital (4, 129, 77)-net over F₁₆	[i]
62	No digital (4, 130, 77)-net over F₁₆	[i]
63	No digital (4, m, 77)-net over F₁₆ for arbitrarily large m	[i]	m-Reduction from Arbitrarily Large Net