MinT - Information on Result #2454246

Information on Result #2454246

Linear OOA(49⁵³, large, F₄₉, 2, 4), using strength reduction based on linear OOA(49⁵³, large, F₄₉, 2, 8), using

generalized (u,Â u+v)-construction [i] based on
1. linear OOA(49⁰, 50, F₄₉, 2, 0) (dual of [(50, 2), 100, 1]-NRT-code), using
  - discarding factors / shortening the dual code based on linear OOA(49⁰, s, F₄₉, 2, 0) for arbitrarily large s, using
    - OOA with only one run / dual of NRT-code without redundancy [i]
2. linear OOA(49⁰, 50, F₄₉, 2, 0) (dual of [(50, 2), 100, 1]-NRT-code) (see above)
3. linear OOA(49⁰, 50, F₄₉, 2, 0) (dual of [(50, 2), 100, 1]-NRT-code) (see above)
4. linear OOA(49⁰, 50, F₄₉, 2, 0) (dual of [(50, 2), 100, 1]-NRT-code) (see above)
5. linear OOA(49⁰, 50, F₄₉, 2, 0) (dual of [(50, 2), 100, 1]-NRT-code) (see above)
6. linear OOA(49⁰, 50, F₄₉, 2, 0) (dual of [(50, 2), 100, 1]-NRT-code) (see above)
7. linear OOA(49⁰, 50, F₄₉, 2, 0) (dual of [(50, 2), 100, 1]-NRT-code) (see above)
8. linear OOA(49⁰, 50, F₄₉, 2, 0) (dual of [(50, 2), 100, 1]-NRT-code) (see above)
9. linear OOA(49⁰, 50, F₄₉, 2, 0) (dual of [(50, 2), 100, 1]-NRT-code) (see above)
10. linear OOA(49⁰, 50, F₄₉, 2, 0) (dual of [(50, 2), 100, 1]-NRT-code) (see above)
11. linear OOA(49⁰, 50, F₄₉, 2, 0) (dual of [(50, 2), 100, 1]-NRT-code) (see above)
12. linear OOA(49⁰, 50, F₄₉, 2, 0) (dual of [(50, 2), 100, 1]-NRT-code) (see above)
13. linear OOA(49⁰, 50, F₄₉, 2, 0) (dual of [(50, 2), 100, 1]-NRT-code) (see above)
14. linear OOA(49⁰, 50, F₄₉, 2, 0) (dual of [(50, 2), 100, 1]-NRT-code) (see above)
15. linear OOA(49⁰, 50, F₄₉, 2, 0) (dual of [(50, 2), 100, 1]-NRT-code) (see above)
16. linear OOA(49⁰, 50, F₄₉, 2, 0) (dual of [(50, 2), 100, 1]-NRT-code) (see above)
17. linear OOA(49⁰, 50, F₄₉, 2, 0) (dual of [(50, 2), 100, 1]-NRT-code) (see above)
18. linear OOA(49⁰, 50, F₄₉, 2, 0) (dual of [(50, 2), 100, 1]-NRT-code) (see above)
19. linear OOA(49⁰, 50, F₄₉, 2, 0) (dual of [(50, 2), 100, 1]-NRT-code) (see above)
20. linear OOA(49⁰, 50, F₄₉, 2, 0) (dual of [(50, 2), 100, 1]-NRT-code) (see above)
21. linear OOA(49⁰, 50, F₄₉, 2, 0) (dual of [(50, 2), 100, 1]-NRT-code) (see above)
22. linear OOA(49⁰, 50, F₄₉, 2, 0) (dual of [(50, 2), 100, 1]-NRT-code) (see above)
23. linear OOA(49⁰, 50, F₄₉, 2, 0) (dual of [(50, 2), 100, 1]-NRT-code) (see above)
24. linear OOA(49⁰, 50, F₄₉, 2, 0) (dual of [(50, 2), 100, 1]-NRT-code) (see above)
25. linear OOA(49⁰, 50, F₄₉, 2, 0) (dual of [(50, 2), 100, 1]-NRT-code) (see above)
26. linear OOA(49⁰, 50, F₄₉, 2, 0) (dual of [(50, 2), 100, 1]-NRT-code) (see above)
27. linear OOA(49⁰, 50, F₄₉, 2, 0) (dual of [(50, 2), 100, 1]-NRT-code) (see above)
28. linear OOA(49⁰, 50, F₄₉, 2, 0) (dual of [(50, 2), 100, 1]-NRT-code) (see above)
29. linear OOA(49⁰, 50, F₄₉, 2, 0) (dual of [(50, 2), 100, 1]-NRT-code) (see above)
30. linear OOA(49⁰, 50, F₄₉, 2, 0) (dual of [(50, 2), 100, 1]-NRT-code) (see above)
31. linear OOA(49⁰, 50, F₄₉, 2, 0) (dual of [(50, 2), 100, 1]-NRT-code) (see above)
32. linear OOA(49⁰, 50, F₄₉, 2, 0) (dual of [(50, 2), 100, 1]-NRT-code) (see above)
33. linear OOA(49⁰, 50, F₄₉, 2, 0) (dual of [(50, 2), 100, 1]-NRT-code) (see above)
34. linear OOA(49⁰, 50, F₄₉, 2, 0) (dual of [(50, 2), 100, 1]-NRT-code) (see above)
35. linear OOA(49⁰, 50, F₄₉, 2, 0) (dual of [(50, 2), 100, 1]-NRT-code) (see above)
36. linear OOA(49⁰, 50, F₄₉, 2, 0) (dual of [(50, 2), 100, 1]-NRT-code) (see above)
37. linear OOA(49⁰, 50, F₄₉, 2, 0) (dual of [(50, 2), 100, 1]-NRT-code) (see above)
38. linear OOA(49⁰, 50, F₄₉, 2, 0) (dual of [(50, 2), 100, 1]-NRT-code) (see above)
39. linear OOA(49⁰, 50, F₄₉, 2, 0) (dual of [(50, 2), 100, 1]-NRT-code) (see above)
40. linear OOA(49⁰, 50, F₄₉, 2, 0) (dual of [(50, 2), 100, 1]-NRT-code) (see above)
41. linear OOA(49⁰, 50, F₄₉, 2, 0) (dual of [(50, 2), 100, 1]-NRT-code) (see above)
42. linear OOA(49¹, 50, F₄₉, 2, 1) (dual of [(50, 2), 99, 2]-NRT-code), using
  - discarding factors / shortening the dual code based on linear OOA(49¹, s, F₄₉, 2, 1) for arbitrarily large s, using
    - appending 1 arbitrary column [i] based on linear OA(49¹, s, F₄₉, 1) (dual of [s, s−1, 2]-code) for arbitrarily large s, using
      - parity-check code [i]
43. linear OOA(49¹, 50, F₄₉, 2, 1) (dual of [(50, 2), 99, 2]-NRT-code) (see above)
44. linear OOA(49¹, 50, F₄₉, 2, 1) (dual of [(50, 2), 99, 2]-NRT-code) (see above)
45. linear OOA(49¹, 50, F₄₉, 2, 1) (dual of [(50, 2), 99, 2]-NRT-code) (see above)
46. linear OOA(49², 50, F₄₉, 2, 2) (dual of [(50, 2), 98, 3]-NRT-code), using
  - extended Reedâ€“Solomon NRT-code RS_e(2;98,49) [i]
47. linear OOA(49⁵, 2882402, F₄₉, 2, 2) (dual of [(2882402, 2), 5764799, 3]-NRT-code), using
  - discarding factors / shortening the dual code based on linear OOA(49⁵, 5884901, F₄₉, 2, 2) (dual of [(5884901, 2), 11769797, 3]-NRT-code), using
    - appending kth column [i] based on linear OA(49⁵, 5884901, F₄₉, 2) (dual of [5884901, 5884896, 3]-code), using
      - Hamming code H(5,49) [i]
48. linear OOA(49¹³, 2882402, F₄₉, 2, 4) (dual of [(2882402, 2), 5764791, 5]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(49¹³, 5764804, F₄₉, 4) (dual of [5764804, 5764791, 5]-code), using
    - discarding factors / shortening the dual code based on linear OA(49¹³, 5764805, F₄₉, 4) (dual of [5764805, 5764792, 5]-code), using
      - constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
        linear OA(49¹³, 5764801, F₄₉, 4) (dual of [5764801, 5764788, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(49⁹, 5764801, F₄₉, 3) (dual of [5764801, 5764792, 4]-code or 5764801-cap in PG(8,49)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
        linear OA(49⁰, 4, F₄₉, 0) (dual of [4, 4, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(49⁰, s, F₄₉, 0) (dual of [s, s, 1]-code) for arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
49. linear OOA(49²⁹, 2882402, F₄₉, 2, 8) (dual of [(2882402, 2), 5764775, 9]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(49²⁹, 5764804, F₄₉, 8) (dual of [5764804, 5764775, 9]-code), using
    - discarding factors / shortening the dual code based on linear OA(49²⁹, 5764805, F₄₉, 8) (dual of [5764805, 5764776, 9]-code), using
      - constructionÂ X applied to C_e(7) âŠ‚ C_e(6) [i] based on
        linear OA(49²⁹, 5764801, F₄₉, 8) (dual of [5764801, 5764772, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
        linear OA(49²⁵, 5764801, F₄₉, 7) (dual of [5764801, 5764776, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(49⁰, 4, F₄₉, 0) (dual of [4, 4, 1]-code) (see above)

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(49⁵⁴, large, F₄₉, 2, 4)	[i]		OOA Duplication