MinT - Information on Result #3171146

Information on Result #3171146

Digital (90, 95, large)-net over F₇, using 3 times m-reduction based on digital (90, 98, large)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (0, 4, 8)-net over F₇, using
  - net from sequence [i] based on digital (0, 7)-sequence over F₇, using
    - Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₇ with g(F) = 0 and N(F) ≥ 8, using
      - the rational function field F₇(x) [i]
    - Niederreiter sequence [i]
2. digital (86, 94, 8388602)-net over F₇, using
  - trace code for nets [i] based on digital (39, 47, 4194301)-net over F₄₉, using
    - net defined by OOA [i] based on linear OOA(49⁴⁷, 4194301, F₄₉, 10, 8) (dual of [(4194301, 10), 41942963, 9]-NRT-code), using
      - OOA 2-folding and stacking with additional row [i] based on linear OOA(49⁴⁷, large, F₄₉, 2, 8), using
        generalized (u,Â u+v)-construction [i] based on
        linear OOA(49⁵, 2882402, F₄₉, 2, 2) (dual of [(2882402, 2), 5764799, 3]-NRT-code), using
        discarding factors / shortening the dual code 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]
        linear OOA(49¹³, 2882402, F₄₉, 2, 4) (dual of [(2882402, 2), 5764791, 5]-NRT-code), using
        OOA 2-folding [i] based on linear OA(49¹³, 5764804, F₄₉, 4) (dual of [5764804, 5764791, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(49¹³, 5764805, F₄₉, 4) (dual of [5764805, 5764792, 5]-code), using
        constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
        linear OA(49¹³, 5764801, F₄₉, 4) (dual of [5764801, 5764788, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(49⁹, 5764801, F₄₉, 3) (dual of [5764801, 5764792, 4]-code or 5764801-cap in PG(8,49)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [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]
        linear OOA(49²⁹, 2882402, F₄₉, 2, 8) (dual of [(2882402, 2), 5764775, 9]-NRT-code), using
        OOA 2-folding [i] based on linear OA(49²⁹, 5764804, F₄₉, 8) (dual of [5764804, 5764775, 9]-code), using
        discarding factors / shortening the dual code based on linear OA(49²⁹, 5764805, F₄₉, 8) (dual of [5764805, 5764776, 9]-code), using
        constructionÂ X applied to C_e(7) âŠ‚ C_e(6) [i] based on
        linear OA(49²⁹, 5764801, F₄₉, 8) (dual of [5764801, 5764772, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
        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(49⁰, 4, F₄₉, 0) (dual of [4, 4, 1]-code) (see above)

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		This result only	Method
1	Digital (91, 95, large)-net over F₇	[i]		t-Expansion
2	Digital (92, 95, large)-net over F₇	[i]		t-Expansion