MinT - Information on Result #1226383

Information on Result #1226383

Digital (34, 40, large)-net over F₄₉, using (u,Â u+v)-construction based on

digital (4, 7, 5769603)-net over F₄₉, using
- net defined by OOA [i] based on linear OOA(49⁷, 5769603, F₄₉, 3, 3) (dual of [(5769603, 3), 17308802, 4]-NRT-code), using
  - appending kth column [i] based on linear OOA(49⁷, 5769603, F₄₉, 2, 3) (dual of [(5769603, 2), 11539199, 4]-NRT-code), using
    - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [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]
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)) (see above)
      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 (34, 39, large)-net over F₄₉	[i]	m-Reduction
2	Digital (34, 38, large)-net over F₄₉	[i]
3	Digital (34, 37, large)-net over F₄₉	[i]