MinT - Information on Result #3226058

Information on Result #3226058

Digital (103, 107, large)-net over F₇, using t-expansion based on digital (101, 107, large)-net over F₇, using

3 times m-reduction [i] based on digital (101, 110, 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 (97, 106, 8388602)-net over F₇, using
    - trace code for nets [i] based on digital (44, 53, 4194301)-net over F₄₉, using
      - net defined by OOA [i] based on linear OOA(49⁵³, 4194301, F₄₉, 10, 9) (dual of [(4194301, 10), 41942957, 10]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OOA(49⁵³, large, F₄₉, 2, 9), using
        generalized (u,Â u+v)-construction [i] based on
        linear OOA(49⁷, 2882402, F₄₉, 2, 3) (dual of [(2882402, 2), 5764797, 4]-NRT-code), using
        discarding factors / shortening the dual code 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]
        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(49³³, 2882402, F₄₉, 2, 9) (dual of [(2882402, 2), 5764771, 10]-NRT-code), using
        OOA 2-folding [i] based on linear OA(49³³, 5764804, F₄₉, 9) (dual of [5764804, 5764771, 10]-code), using
        discarding factors / shortening the dual code based on linear OA(49³³, 5764805, F₄₉, 9) (dual of [5764805, 5764772, 10]-code), using
        constructionÂ X applied to C_e(8) âŠ‚ C_e(7) [i] based on
        linear OA(49³³, 5764801, F₄₉, 9) (dual of [5764801, 5764768, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
        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(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

None.