MinT - Information on Result #3177246

Information on Result #3177246

Digital (75, 82, large)-net over F₂₇, using 3 times m-reduction based on digital (75, 85, large)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 0, 265722)-net over F₂₇, using
  - s-reduction based on digital (0, 0, s)-net over F₂₇ for arbitrarily large s, using
    - arbitrary set of 27⁰ points [i]
2. digital (0, 0, 265722)-net over F₂₇ (see above)
3. digital (0, 0, 265722)-net over F₂₇ (see above)
4. digital (0, 0, 265722)-net over F₂₇ (see above)
5. digital (0, 0, 265722)-net over F₂₇ (see above)
6. digital (0, 0, 265722)-net over F₂₇ (see above)
7. digital (0, 0, 265722)-net over F₂₇ (see above)
8. digital (0, 0, 265722)-net over F₂₇ (see above)
9. digital (0, 0, 265722)-net over F₂₇ (see above)
10. digital (0, 0, 265722)-net over F₂₇ (see above)
11. digital (0, 0, 265722)-net over F₂₇ (see above)
12. digital (0, 0, 265722)-net over F₂₇ (see above)
13. digital (0, 0, 265722)-net over F₂₇ (see above)
14. digital (0, 0, 265722)-net over F₂₇ (see above)
15. digital (0, 0, 265722)-net over F₂₇ (see above)
16. digital (0, 0, 265722)-net over F₂₇ (see above)
17. digital (0, 0, 265722)-net over F₂₇ (see above)
18. digital (0, 1, 265722)-net over F₂₇, using
  - s-reduction based on digital (0, 1, s)-net over F₂₇ for arbitrarily large s, using
    - construction for strength kÂ =Â 1 [i]
19. digital (0, 1, 265722)-net over F₂₇ (see above)
20. digital (0, 1, 265722)-net over F₂₇ (see above)
21. digital (0, 1, 265722)-net over F₂₇ (see above)
22. digital (0, 1, 265722)-net over F₂₇ (see above)
23. digital (3, 5, 265722)-net over F₂₇, using
  - s-reduction based on digital (3, 5, 551881)-net over F₂₇, using
    - kÂ =Â 2 construction [i]
24. digital (3, 5, 265722)-net over F₂₇ (see above)
25. digital (4, 7, 265722)-net over F₂₇, using
  - s-reduction based on digital (4, 7, 532899)-net over F₂₇, using
    - net defined by OOA [i] based on linear OOA(27⁷, 532899, F₂₇, 3, 3) (dual of [(532899, 3), 1598690, 4]-NRT-code), using
      - appending kth column [i] based on linear OOA(27⁷, 532899, F₂₇, 2, 3) (dual of [(532899, 2), 1065791, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(27⁷, 532899, F₂₇, 3) (dual of [532899, 532892, 4]-code or 532899-cap in PG(6,27)), using
        product of two caps with tangent hyperplane [i]
26. digital (12, 17, 265722)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27¹⁷, 265722, F₂₇, 5, 5) (dual of [(265722, 5), 1328593, 6]-NRT-code), using
    - OOA 2-folding and stacking with additional row [i] based on linear OA(27¹⁷, 531445, F₂₇, 5) (dual of [531445, 531428, 6]-code), using
      - constructionÂ X applied to C_e(4) âŠ‚ C_e(3) [i] based on
        linear OA(27¹⁷, 531441, F₂₇, 5) (dual of [531441, 531424, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 27⁴âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(27¹³, 531441, F₂₇, 4) (dual of [531441, 531428, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 27⁴âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(27⁰, 4, F₂₇, 0) (dual of [4, 4, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(27⁰, s, F₂₇, 0) (dual of [s, s, 1]-code) for arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
27. digital (36, 46, 1677720)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27⁴⁶, 1677720, F₂₇, 10, 10) (dual of [(1677720, 10), 16777154, 11]-NRT-code), using
    - OA 5-folding and stacking [i] based on linear OA(27⁴⁶, 8388600, F₂₇, 10) (dual of [8388600, 8388554, 11]-code), using
      - discarding factors / shortening the dual code based on linear OA(27⁴⁶, large, F₂₇, 10) (dual of [large, large−46, 11]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 27⁵âˆ’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		Method
1	Digital (76, 82, large)-net over F₂₇	[i]	t-Expansion
2	Digital (77, 82, large)-net over F₂₇	[i]
3	Digital (78, 82, large)-net over F₂₇	[i]
4	Digital (79, 82, large)-net over F₂₇	[i]