MinT - Information on Result #1205381

Information on Result #1205381

Digital (60, 66, large)-net over F₇, using trace code for nets based on digital (27, 33, 5764802)-net over F₄₉, using

net defined by OOA [i] based on linear OOA(49³³, 5764802, F₄₉, 9, 6) (dual of [(5764802, 9), 51883185, 7]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(49³³, 5764803, F₄₉, 3, 6) (dual of [(5764803, 3), 17294376, 7]-NRT-code), using
  - generalized (u,Â u+v)-construction [i] based on
    1. linear OOA(49⁵, 1921601, F₄₉, 3, 2) (dual of [(1921601, 3), 5764798, 3]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(49⁵, 5884901, F₄₉, 3, 2) (dual of [(5884901, 3), 17654698, 3]-NRT-code), using
        appending 1 arbitrary column [i] based on linear OOA(49⁵, 5884901, F₄₉, 2, 2) (dual of [(5884901, 2), 11769797, 3]-NRT-code), using
        appending kth column [i] based on linear OA(49⁵, 5884901, F₄₉, 2) (dual of [5884901, 5884896, 3]-code), using
        Hamming code H(5,49) [i]
    2. linear OOA(49⁷, 1921601, F₄₉, 3, 3) (dual of [(1921601, 3), 5764796, 4]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(49⁷, 5769602, F₄₉, 3, 3) (dual of [(5769602, 3), 17308799, 4]-NRT-code), using
        OOA stacking with additional row [i] based on linear OA(49⁷, 5769603, F₄₉, 3) (dual of [5769603, 5769596, 4]-code or 5769603-cap in PG(6,49)), using
        product of two caps with tangent hyperplane [i]
    3. linear OOA(49²¹, 1921601, F₄₉, 3, 6) (dual of [(1921601, 3), 5764782, 7]-NRT-code), using
      - OOA 3-folding [i] based on linear OA(49²¹, 5764803, F₄₉, 6) (dual of [5764803, 5764782, 7]-code), using
        discarding factors / shortening the dual code based on linear OA(49²¹, 5764805, F₄₉, 6) (dual of [5764805, 5764784, 7]-code), using
        constructionÂ X applied to C_e(5) âŠ‚ C_e(4) [i] based on
        linear OA(49²¹, 5764801, F₄₉, 6) (dual of [5764801, 5764780, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(49¹⁷, 5764801, F₄₉, 5) (dual of [5764801, 5764784, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(49⁰, 4, F₄₉, 0) (dual of [4, 4, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(49⁰, s, F₄₉, 0) (dual of [s, s, 1]-code) for arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

Mode: Constructive and linear.

Optimality

Show details for fixed k and m, k and t, m and t.

Other Results with Identical Parameters

None.

Depending Results

The following results depend on this result:

	Result		Method
1	Digital (61, 66, large)-net over F₇	[i]	t-Expansion
2	Digital (62, 66, large)-net over F₇	[i]
3	Digital (63, 66, large)-net over F₇	[i]
4	Digital (61, 67, large)-net over F₇	[i]	Net Duplication
5	Digital (62, 68, large)-net over F₇	[i]
6	Digital (63, 69, large)-net over F₇	[i]
7	Digital (60, 65, large)-net over F₇	[i]	m-Reduction
8	Digital (60, 64, large)-net over F₇	[i]
9	Digital (60, 63, large)-net over F₇	[i]