Information on Result #700517
Linear OA(227, 78, F2, 8) (dual of [78, 51, 9]-code), using construction XX applied to C1 = C({0,1,3,31}), C2 = C([1,5]), C3 = C1 + C2 = C([1,3]), and C∩ = C1 ∩ C2 = C({0,1,3,5,31}) based on
- linear OA(219, 63, F2, 7) (dual of [63, 44, 8]-code), using the primitive cyclic code C(A) with length 63 = 26−1, defining set A = {0,1,3,31}, and minimum distance d ≥ |{−2,−1,…,4}|+1 = 8 (BCH-bound) [i]
- linear OA(218, 63, F2, 6) (dual of [63, 45, 7]-code), using the primitive narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(225, 63, F2, 9) (dual of [63, 38, 10]-code), using the primitive cyclic code C(A) with length 63 = 26−1, defining set A = {0,1,3,5,31}, and minimum distance d ≥ |{−2,−1,…,6}|+1 = 10 (BCH-bound) [i]
- linear OA(212, 63, F2, 4) (dual of [63, 51, 5]-code), using the primitive narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(21, 8, F2, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) for arbitrarily large s, using
- linear OA(21, 7, F2, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) for arbitrarily large s (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.
Compare with Markus Grassl’s online database of code parameters.
Other Results with Identical Parameters
None.
Depending Results
The following results depend on this result:
| Result | This result only | Method | ||
|---|---|---|---|---|
| 1 | Linear OOA(227, 54, F2, 2, 8) (dual of [(54, 2), 81, 9]-NRT-code) | [i] | Embedding of OOA with Gilbert–Varšamov Bound | |
| 2 | Linear OOA(227, 54, F2, 3, 8) (dual of [(54, 3), 135, 9]-NRT-code) | [i] | ||
| 3 | Linear OOA(227, 54, F2, 4, 8) (dual of [(54, 4), 189, 9]-NRT-code) | [i] | ||
| 4 | Linear OOA(227, 54, F2, 5, 8) (dual of [(54, 5), 243, 9]-NRT-code) | [i] | ||
| 5 | Linear OOA(227, 54, F2, 6, 8) (dual of [(54, 6), 297, 9]-NRT-code) | [i] | ||
| 6 | Linear OOA(227, 54, F2, 7, 8) (dual of [(54, 7), 351, 9]-NRT-code) | [i] | ||
| 7 | Linear OA(228, 79, F2, 9) (dual of [79, 51, 10]-code) | [i] | Adding a Parity Check Bit | |
| 8 | Linear OOA(227, 39, F2, 2, 8) (dual of [(39, 2), 51, 9]-NRT-code) | [i] | OOA Folding | |
| 9 | Linear OOA(227, 26, F2, 3, 8) (dual of [(26, 3), 51, 9]-NRT-code) | [i] | ||