MinT - Information on Result #3176885

Information on Result #3176885

Digital (79, 84, large)-net over F₂₅, using 5 times m-reduction based on digital (79, 89, large)-net over F₂₅, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 0, 391875)-net over F₂₅, using
  - s-reduction based on digital (0, 0, s)-net over F₂₅ for arbitrarily large s, using
    - arbitrary set of 25⁰ points [i]
2. digital (0, 0, 391875)-net over F₂₅ (see above)
3. digital (0, 0, 391875)-net over F₂₅ (see above)
4. digital (0, 0, 391875)-net over F₂₅ (see above)
5. digital (0, 0, 391875)-net over F₂₅ (see above)
6. digital (0, 0, 391875)-net over F₂₅ (see above)
7. digital (0, 0, 391875)-net over F₂₅ (see above)
8. digital (0, 0, 391875)-net over F₂₅ (see above)
9. digital (0, 0, 391875)-net over F₂₅ (see above)
10. digital (0, 0, 391875)-net over F₂₅ (see above)
11. digital (0, 0, 391875)-net over F₂₅ (see above)
12. digital (0, 0, 391875)-net over F₂₅ (see above)
13. digital (0, 0, 391875)-net over F₂₅ (see above)
14. digital (0, 0, 391875)-net over F₂₅ (see above)
15. digital (0, 0, 391875)-net over F₂₅ (see above)
16. digital (0, 1, 391875)-net over F₂₅, using
  - s-reduction based on digital (0, 1, s)-net over F₂₅ for arbitrarily large s, using
    - construction for strength kÂ =Â 1 [i]
17. digital (0, 1, 391875)-net over F₂₅ (see above)
18. digital (0, 1, 391875)-net over F₂₅ (see above)
19. digital (0, 1, 391875)-net over F₂₅ (see above)
20. digital (0, 1, 391875)-net over F₂₅ (see above)
21. digital (3, 5, 391875)-net over F₂₅, using
  - s-reduction based on digital (3, 5, 406901)-net over F₂₅, using
    - kÂ =Â 2 construction [i]
22. digital (3, 5, 391875)-net over F₂₅ (see above)
23. digital (4, 7, 391875)-net over F₂₅, using
  - net defined by OOA [i] based on linear OOA(25⁷, 391875, F₂₅, 3, 3) (dual of [(391875, 3), 1175618, 4]-NRT-code), using
    - appending kth column [i] based on linear OOA(25⁷, 391875, F₂₅, 2, 3) (dual of [(391875, 2), 783743, 4]-NRT-code), using
      - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(25⁷, 391875, F₂₅, 3) (dual of [391875, 391868, 4]-code or 391875-cap in PG(6,25)), using
        product of two caps with tangent hyperplane [i]
24. digital (16, 21, 1677720)-net over F₂₅, using
  - s-reduction based on digital (16, 21, 4194301)-net over F₂₅, using
    - net defined by OOA [i] based on linear OOA(25²¹, 4194301, F₂₅, 5, 5) (dual of [(4194301, 5), 20971484, 6]-NRT-code), using
      - OOA 2-folding and stacking with additional row [i] based on linear OA(25²¹, large, F₂₅, 5) (dual of [large, large−21, 6]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 9765626Â | 25¹⁰âˆ’1, defining interval IÂ =Â [0,2], and minimum distance dÂ â‰¥ |{−2,−1,0,1,2}|+1Â =Â 6 (BCH-bound) [i]
25. digital (36, 46, 1677720)-net over F₂₅, using
  - net defined by OOA [i] based on linear OOA(25⁴⁶, 1677720, F₂₅, 10, 10) (dual of [(1677720, 10), 16777154, 11]-NRT-code), using
    - OA 5-folding and stacking [i] based on linear OA(25⁴⁶, 8388600, F₂₅, 10) (dual of [8388600, 8388554, 11]-code), using
      - discarding factors / shortening the dual code based on linear OA(25⁴⁶, large, F₂₅, 10) (dual of [large, large−46, 11]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 25⁵âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [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		This result only	Method
1	Digital (80, 84, large)-net over F₂₅	[i]		t-Expansion
2	Digital (81, 84, large)-net over F₂₅	[i]		t-Expansion