MinT - Information on Result #1226395

Information on Result #1226395

Digital (35, 42, large)-net over F₄₉, using (u,Â u+v)-construction based on

digital (0, 3, 50)-net over F₄₉, using
- net from sequence [i] based on digital (0, 49)-sequence over F₄₉, using
  - generalized Faure sequence [i]
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄₉ with g(F) = 0 and N(F) ≥ 50, using
    - the rational function field F₄₉(x) [i]
  - Niederreiter sequence [i]
digital (32, 39, 8388602)-net over F₄₉, using
- net defined by OOA [i] based on linear OOA(49³⁹, 8388602, F₄₉, 9, 7) (dual of [(8388602, 9), 75497379, 8]-NRT-code), using
  - OOA stacking with additional row [i] based on linear OOA(49³⁹, large, F₄₉, 3, 7), using
    - generalized (u,Â u+v)-construction [i] based on
      1. linear OOA(49¹, 1921601, F₄₉, 3, 1) (dual of [(1921601, 3), 5764802, 2]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(49¹, s, F₄₉, 3, 1) for arbitrarily large s, using
        appending 2 arbitrary columns [i] based on linear OA(49¹, s, F₄₉, 1) (dual of [s, s−1, 2]-code) for arbitrarily large s, using
        parity-check code [i]
      2. linear OOA(49¹, 1921601, F₄₉, 3, 1) (dual of [(1921601, 3), 5764802, 2]-NRT-code) (see above)
      3. 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]
      4. 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]
      5. linear OOA(49²⁵, 1921601, F₄₉, 3, 7) (dual of [(1921601, 3), 5764778, 8]-NRT-code), using
        OOA 3-folding [i] based on linear OA(49²⁵, 5764803, F₄₉, 7) (dual of [5764803, 5764778, 8]-code), using
        discarding factors / shortening the dual code based on linear OA(49²⁵, 5764805, F₄₉, 7) (dual of [5764805, 5764780, 8]-code), using
        constructionÂ X applied to C_e(6) âŠ‚ C_e(5) [i] based on
        linear OA(49²⁵, 5764801, F₄₉, 7) (dual of [5764801, 5764776, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        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(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 (36, 42, large)-net over F₄₉	[i]	t-Expansion
2	Digital (36, 43, large)-net over F₄₉	[i]	Net Duplication
3	Digital (35, 41, large)-net over F₄₉	[i]	m-Reduction
4	Digital (35, 40, large)-net over F₄₉	[i]
5	Digital (35, 39, large)-net over F₄₉	[i]
6	Digital (35, 38, large)-net over F₄₉	[i]