MinT - Information on Result #2519673

Information on Result #2519673

Linear OOA(32⁴³, large, F₃₂, 2, 7), using 1 times NRT-code embedding in larger space based on linear OOA(32⁴¹, large, F₃₂, 2, 7), using

generalized (u,Â u+v)-construction [i] based on
1. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code), using
  - discarding factors / shortening the dual code based on linear OOA(32⁰, s, F₃₂, 2, 0) for arbitrarily large s, using
    - OOA with only one run / dual of NRT-code without redundancy [i]
2. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
3. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
4. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
5. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
6. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
7. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
8. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
9. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
10. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
11. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
12. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
13. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
14. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
15. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
16. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
17. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
18. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
19. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
20. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
21. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
22. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
23. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
24. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
25. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
26. linear OOA(32¹, 524290, F₃₂, 2, 1) (dual of [(524290, 2), 1048579, 2]-NRT-code), using
  - discarding factors / shortening the dual code based on linear OOA(32¹, s, F₃₂, 2, 1) for arbitrarily large s, using
    - appending 1 arbitrary column [i] based on linear OA(32¹, s, F₃₂, 1) (dual of [s, s−1, 2]-code) for arbitrarily large s, using
      - parity-check code [i]
27. linear OOA(32¹, 524290, F₃₂, 2, 1) (dual of [(524290, 2), 1048579, 2]-NRT-code) (see above)
28. linear OOA(32¹, 524290, F₃₂, 2, 1) (dual of [(524290, 2), 1048579, 2]-NRT-code) (see above)
29. linear OOA(32¹, 524290, F₃₂, 2, 1) (dual of [(524290, 2), 1048579, 2]-NRT-code) (see above)
30. linear OOA(32⁵, 524290, F₃₂, 2, 2) (dual of [(524290, 2), 1048575, 3]-NRT-code), using
  - discarding factors / shortening the dual code based on linear OOA(32⁵, 1082401, F₃₂, 2, 2) (dual of [(1082401, 2), 2164797, 3]-NRT-code), using
    - appending kth column [i] based on linear OA(32⁵, 1082401, F₃₂, 2) (dual of [1082401, 1082396, 3]-code), using
      - Hamming code H(5,32) [i]
31. linear OOA(32⁷, 524290, F₃₂, 2, 3) (dual of [(524290, 2), 1048573, 4]-NRT-code), using
  - discarding factors / shortening the dual code 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]
32. linear OOA(32²⁵, 524290, F₃₂, 2, 7) (dual of [(524290, 2), 1048555, 8]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(32²⁵, 1048580, F₃₂, 7) (dual of [1048580, 1048555, 8]-code), using
    - constructionÂ X applied to C_e(6) âŠ‚ C_e(5) [i] based on
      1. linear OA(32²⁵, 1048576, F₃₂, 7) (dual of [1048576, 1048551, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
      2. linear OA(32²¹, 1048576, F₃₂, 6) (dual of [1048576, 1048555, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
      3. 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]

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	Linear OOA(32⁴⁴, large, F₃₂, 2, 7)	[i]		OOA Duplication