MinT - Information on Result #1234578

Information on Result #1234578

Digital (41, 50, large)-net over F₁₂₈, using generalized (u,Â u+v)-construction based on

digital (0, 0, 524288)-net over F₁₂₈, using
- s-reduction based on digital (0, 0, s)-net over F₁₂₈ for arbitrarily large s, using
  - arbitrary set of 128⁰ points [i]
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 0, 524288)-net over F₁₂₈ (see above)
digital (0, 1, 524288)-net over F₁₂₈, using
- s-reduction based on digital (0, 1, s)-net over F₁₂₈ for arbitrarily large s, using
  - construction for strength kÂ =Â 1 [i]
digital (0, 1, 524288)-net over F₁₂₈ (see above)
digital (0, 1, 524288)-net over F₁₂₈ (see above)
digital (0, 1, 524288)-net over F₁₂₈ (see above)
digital (0, 1, 524288)-net over F₁₂₈ (see above)
digital (2, 4, 524288)-net over F₁₂₈, using
- s-reduction based on digital (2, 4, 2113665)-net over F₁₂₈, using
  - kÂ =Â 2 construction [i]
digital (3, 6, 524288)-net over F₁₂₈, using
- s-reduction based on digital (3, 6, 2130050)-net over F₁₂₈, using
  - net defined by OOA [i] based on linear OOA(128⁶, 2130050, F₁₂₈, 3, 3) (dual of [(2130050, 3), 6390144, 4]-NRT-code), using
    - appending kth column [i] based on linear OOA(128⁶, 2130050, F₁₂₈, 2, 3) (dual of [(2130050, 2), 4260094, 4]-NRT-code), using
      - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(128⁶, 2130050, F₁₂₈, 3) (dual of [2130050, 2130044, 4]-code or 2130050-cap in PG(5,128)), using
        large caps in PG(5,Â b) [i]
digital (6, 10, 524288)-net over F₁₂₈, using
- s-reduction based on digital (6, 10, 1048577)-net over F₁₂₈, using
  - net defined by OOA [i] based on linear OOA(128¹⁰, 1048577, F₁₂₈, 4, 4) (dual of [(1048577, 4), 4194298, 5]-NRT-code), using
    - OA 2-folding and stacking [i] based on linear OA(128¹⁰, 2097154, F₁₂₈, 4) (dual of [2097154, 2097144, 5]-code), using
      - discarding factors / shortening the dual code based on linear OA(128¹⁰, 2097155, F₁₂₈, 4) (dual of [2097155, 2097145, 5]-code), using
        constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
        linear OA(128¹⁰, 2097152, F₁₂₈, 4) (dual of [2097152, 2097142, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(128⁷, 2097152, F₁₂₈, 3) (dual of [2097152, 2097145, 4]-code or 2097152-cap in PG(6,128)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
        linear OA(128⁰, 3, F₁₂₈, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(128⁰, s, F₁₂₈, 0) (dual of [s, s, 1]-code) for arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
digital (16, 25, 524288)-net over F₁₂₈, using
- net defined by OOA [i] based on linear OOA(128²⁵, 524288, F₁₂₈, 9, 9) (dual of [(524288, 9), 4718567, 10]-NRT-code), using
  - OOA 4-folding and stacking with additional row [i] based on linear OA(128²⁵, 2097153, F₁₂₈, 9) (dual of [2097153, 2097128, 10]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 2097153Â | 128⁶âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [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 (42, 50, large)-net over F₁₂₈	[i]	t-Expansion
2	Digital (43, 50, large)-net over F₁₂₈	[i]
3	Digital (44, 50, large)-net over F₁₂₈	[i]
4	Digital (42, 51, large)-net over F₁₂₈	[i]	Net Duplication
5	Digital (43, 52, large)-net over F₁₂₈	[i]
6	Digital (44, 53, large)-net over F₁₂₈	[i]
7	Digital (41, 49, large)-net over F₁₂₈	[i]	m-Reduction
8	Digital (41, 48, large)-net over F₁₂₈	[i]
9	Digital (41, 47, large)-net over F₁₂₈	[i]
10	Digital (41, 46, large)-net over F₁₂₈	[i]
11	Digital (41, 45, large)-net over F₁₂₈	[i]
12	Digital (41, 44, large)-net over F₁₂₈	[i]