MinT - Information on Result #1233705

Information on Result #1233705

Digital (94, 108, 2397768)-net over F₃₂, using generalized (u,Â u+v)-construction based on

digital (1, 4, 513)-net over F₃₂, using
- s-reduction based on digital (1, 4, 1025)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁴, 1025, F₃₂, 3, 3) (dual of [(1025, 3), 3071, 4]-NRT-code), using
    - appending kth column [i] based on linear OOA(32⁴, 1025, F₃₂, 2, 3) (dual of [(1025, 2), 2046, 4]-NRT-code), using
      - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(32⁴, 1025, F₃₂, 3) (dual of [1025, 1021, 4]-code or 1025-cap in PG(3,32)), using
        ovoid in PG(3,Â 32) [i]
digital (3, 7, 513)-net over F₃₂, using
- net defined by OOA [i] based on linear OOA(32⁷, 513, F₃₂, 4, 4) (dual of [(513, 4), 2045, 5]-NRT-code), using
  - OA 2-folding and stacking [i] based on linear OA(32⁷, 1026, F₃₂, 4) (dual of [1026, 1019, 5]-code), using
    - constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
      1. linear OA(32⁷, 1024, F₃₂, 4) (dual of [1024, 1017, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
      2. linear OA(32⁵, 1024, F₃₂, 3) (dual of [1024, 1019, 4]-code or 1024-cap in PG(4,32)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
      3. linear OA(32⁰, 2, F₃₂, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(32⁰, s, F₃₂, 0) (dual of [s, s, 1]-code) for arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
digital (24, 31, 1198371)-net over F₃₂, using
- s-reduction based on digital (24, 31, 2796200)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32³¹, 2796200, F₃₂, 7, 7) (dual of [(2796200, 7), 19573369, 8]-NRT-code), using
    - OOA 3-folding and stacking with additional row [i] based on linear OA(32³¹, 8388601, F₃₂, 7) (dual of [8388601, 8388570, 8]-code), using
      - discarding factors / shortening the dual code based on linear OA(32³¹, large, F₃₂, 7) (dual of [large, large−31, 8]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 33554433Â | 32¹⁰âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]
digital (52, 66, 1198371)-net over F₃₂, using
- net defined by OOA [i] based on linear OOA(32⁶⁶, 1198371, F₃₂, 14, 14) (dual of [(1198371, 14), 16777128, 15]-NRT-code), using
  - OA 7-folding and stacking [i] based on linear OA(32⁶⁶, 8388597, F₃₂, 14) (dual of [8388597, 8388531, 15]-code), using
    - discarding factors / shortening the dual code based on linear OA(32⁶⁶, large, F₃₂, 14) (dual of [large, large−66, 15]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 33554431Â = 32⁵âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

Mode: Constructive and linear.

Optimality

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

Other Results with Identical Parameters

None.

Depending Results

None.