MinT - Information on Result #864938

Information on Result #864938

Linear OOA(27¹², 9882, F₂₇, 2, 4) (dual of [(9882, 2), 19752, 5]-NRT-code), using OOA 2-folding based on linear OA(27¹², 19764, F₂₇, 4) (dual of [19764, 19752, 5]-code), using

generalized (u,Â u+v)-construction [i] based on
1. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 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]
2. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code) (see above)
3. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code) (see above)
4. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code) (see above)
5. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code) (see above)
6. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code) (see above)
7. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code) (see above)
8. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code) (see above)
9. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code) (see above)
10. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code) (see above)
11. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code) (see above)
12. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code) (see above)
13. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code) (see above)
14. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code) (see above)
15. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code) (see above)
16. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code) (see above)
17. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code) (see above)
18. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code) (see above)
19. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code) (see above)
20. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code) (see above)
21. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code) (see above)
22. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code) (see above)
23. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code) (see above)
24. linear OA(27¹, 732, F₂₇, 1) (dual of [732, 731, 2]-code), using
  - discarding factors / shortening the dual code based on linear OA(27¹, s, F₂₇, 1) (dual of [s, s−1, 2]-code) for arbitrarily large s, using
    - parity-check code [i]
25. linear OA(27¹, 732, F₂₇, 1) (dual of [732, 731, 2]-code) (see above)
26. linear OA(27³, 732, F₂₇, 2) (dual of [732, 729, 3]-code), using
  - discarding factors / shortening the dual code based on linear OA(27³, 757, F₂₇, 2) (dual of [757, 754, 3]-code), using
    - Hamming code H(3,27) [i]
27. linear OA(27⁷, 732, F₂₇, 4) (dual of [732, 725, 5]-code), using
  - constructionÂ XX applied to C₁Â =Â C([727,1]), C₂Â =Â C([0,2]), C₃Â =Â C₁Â +Â C₂Â =Â C([0,1]), and C_âˆ©Â =Â C₁Â âˆ©Â C₂Â =Â C([727,2]) [i] based on
    1. linear OA(27⁵, 728, F₂₇, 3) (dual of [728, 723, 4]-code or 728-cap in PG(4,27)), using the primitive BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â {−1,0,1}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
    2. linear OA(27⁵, 728, F₂₇, 3) (dual of [728, 723, 4]-code or 728-cap in PG(4,27)), using the primitive expurgated narrow-sense BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â [0,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
    3. linear OA(27⁷, 728, F₂₇, 4) (dual of [728, 721, 5]-code), using the primitive BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â {−1,0,1,2}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
    4. linear OA(27³, 728, F₂₇, 2) (dual of [728, 725, 3]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â [0,1], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
    5. linear OA(27⁰, 2, F₂₇, 0) (dual of [2, 2, 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 (see above)
    6. linear OA(27⁰, 2, F₂₇, 0) (dual of [2, 2, 1]-code) (see above)

Mode: Constructive and linear.

Optimality

Show details for fixed k and m, n and k, k and s, k and t, n and m, m and s, m and t, n and s, n 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(27⁴⁸, 4204183, F₂₇, 2, 8) (dual of [(4204183, 2), 8408318, 9]-NRT-code)	[i]		(u,Â u+v)-Construction for OOAs
2	Linear OOA(27⁵³, 4204183, F₂₇, 2, 9) (dual of [(4204183, 2), 8408313, 10]-NRT-code)	[i]		(u,Â u+v)-Construction for OOAs