MinT - Information on Result #1233001

Information on Result #1233001

Digital (80, 91, large)-net over F₃₂, using generalized (u,Â u+v)-construction based on

digital (0, 0, 524289)-net over F₃₂, using
- s-reduction based on digital (0, 0, s)-net over F₃₂ for arbitrarily large s, using
  - arbitrary set of 32⁰ points [i]
digital (0, 0, 524289)-net over F₃₂ (see above)
digital (0, 0, 524289)-net over F₃₂ (see above)
digital (0, 0, 524289)-net over F₃₂ (see above)
digital (0, 0, 524289)-net over F₃₂ (see above)
digital (0, 0, 524289)-net over F₃₂ (see above)
digital (0, 0, 524289)-net over F₃₂ (see above)
digital (0, 0, 524289)-net over F₃₂ (see above)
digital (0, 0, 524289)-net over F₃₂ (see above)
digital (0, 0, 524289)-net over F₃₂ (see above)
digital (0, 0, 524289)-net over F₃₂ (see above)
digital (0, 0, 524289)-net over F₃₂ (see above)
digital (0, 0, 524289)-net over F₃₂ (see above)
digital (0, 0, 524289)-net over F₃₂ (see above)
digital (0, 0, 524289)-net over F₃₂ (see above)
digital (0, 0, 524289)-net over F₃₂ (see above)
digital (0, 0, 524289)-net over F₃₂ (see above)
digital (0, 0, 524289)-net over F₃₂ (see above)
digital (0, 0, 524289)-net over F₃₂ (see above)
digital (0, 0, 524289)-net over F₃₂ (see above)
digital (0, 0, 524289)-net over F₃₂ (see above)
digital (0, 1, 524289)-net over F₃₂, using
- s-reduction based on digital (0, 1, s)-net over F₃₂ for arbitrarily large s, using
  - construction for strength kÂ =Â 1 [i]
digital (0, 1, 524289)-net over F₃₂ (see above)
digital (0, 1, 524289)-net over F₃₂ (see above)
digital (0, 1, 524289)-net over F₃₂ (see above)
digital (0, 1, 524289)-net over F₃₂ (see above)
digital (0, 1, 524289)-net over F₃₂ (see above)
digital (3, 5, 524289)-net over F₃₂, using
- s-reduction based on digital (3, 5, 1082401)-net over F₃₂, using
  - kÂ =Â 2 construction [i]
digital (3, 5, 524289)-net over F₃₂ (see above)
digital (4, 7, 524289)-net over F₃₂, using
- s-reduction based on digital (4, 7, 1050624)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁷, 1050624, F₃₂, 3, 3) (dual of [(1050624, 3), 3151865, 4]-NRT-code), using
    - appending kth column [i] based on linear OOA(32⁷, 1050624, F₃₂, 2, 3) (dual of [(1050624, 2), 2101241, 4]-NRT-code), using
      - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(32⁷, 1050624, F₃₂, 3) (dual of [1050624, 1050617, 4]-code or 1050624-cap in PG(6,32)), using
        product of two caps with tangent hyperplane [i]
digital (12, 17, 524289)-net over F₃₂, using
- net defined by OOA [i] based on linear OOA(32¹⁷, 524289, F₃₂, 5, 5) (dual of [(524289, 5), 2621428, 6]-NRT-code), using
  - OOA 2-folding and stacking with additional row [i] based on linear OA(32¹⁷, 1048579, F₃₂, 5) (dual of [1048579, 1048562, 6]-code), using
    - discarding factors / shortening the dual code based on linear OA(32¹⁷, 1048580, F₃₂, 5) (dual of [1048580, 1048563, 6]-code), using
      - constructionÂ X applied to C_e(4) âŠ‚ C_e(3) [i] based on
        linear OA(32¹⁷, 1048576, F₃₂, 5) (dual of [1048576, 1048559, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(32¹³, 1048576, F₃₂, 4) (dual of [1048576, 1048563, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(32⁰, 4, F₃₂, 0) (dual of [4, 4, 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 (40, 51, 1677720)-net over F₃₂, using
- net defined by OOA [i] based on linear OOA(32⁵¹, 1677720, F₃₂, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
  - OOA 5-folding and stacking with additional row [i] based on linear OA(32⁵¹, 8388601, F₃₂, 11) (dual of [8388601, 8388550, 12]-code), using
    - discarding factors / shortening the dual code based on linear OA(32⁵¹, large, F₃₂, 11) (dual of [large, large−51, 12]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 33554433Â | 32¹⁰âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (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 (81, 91, large)-net over F₃₂	[i]	t-Expansion
2	Digital (82, 91, large)-net over F₃₂	[i]
3	Digital (83, 91, large)-net over F₃₂	[i]
4	Digital (84, 91, large)-net over F₃₂	[i]
5	Digital (85, 91, large)-net over F₃₂	[i]
6	Digital (86, 91, large)-net over F₃₂	[i]
7	Digital (87, 91, large)-net over F₃₂	[i]
8	Digital (88, 91, large)-net over F₃₂	[i]
9	Digital (81, 92, large)-net over F₃₂	[i]	Net Duplication
10	Digital (82, 93, large)-net over F₃₂	[i]
11	Digital (83, 94, large)-net over F₃₂	[i]
12	Digital (84, 95, large)-net over F₃₂	[i]
13	Digital (85, 96, large)-net over F₃₂	[i]
14	Digital (86, 97, large)-net over F₃₂	[i]
15	Digital (87, 98, large)-net over F₃₂	[i]
16	Digital (88, 99, large)-net over F₃₂	[i]
17	Digital (89, 100, large)-net over F₃₂	[i]
18	Digital (90, 101, large)-net over F₃₂	[i]
19	Digital (91, 102, large)-net over F₃₂	[i]
20	Digital (92, 103, large)-net over F₃₂	[i]
21	Digital (93, 104, large)-net over F₃₂	[i]
22	Digital (94, 105, large)-net over F₃₂	[i]
23	Digital (95, 106, large)-net over F₃₂	[i]
24	Digital (96, 107, large)-net over F₃₂	[i]
25	Digital (97, 108, large)-net over F₃₂	[i]
26	Digital (98, 109, large)-net over F₃₂	[i]
27	Digital (80, 90, large)-net over F₃₂	[i]	m-Reduction
28	Digital (80, 89, large)-net over F₃₂	[i]
29	Digital (80, 88, large)-net over F₃₂	[i]
30	Digital (80, 87, large)-net over F₃₂	[i]
31	Digital (80, 86, large)-net over F₃₂	[i]
32	Digital (80, 85, large)-net over F₃₂	[i]
33	Digital (80, 84, large)-net over F₃₂	[i]
34	Digital (80, 83, large)-net over F₃₂	[i]