Information on Result #700973
Linear OA(414, 70, F4, 6) (dual of [70, 56, 7]-code), using construction XX applied to C1 = C({0,1,2,47}), C2 = C([0,3]), C3 = C1 + C2 = C([0,2]), and C∩ = C1 ∩ C2 = C({0,1,2,3,47}) based on
- linear OA(410, 63, F4, 4) (dual of [63, 53, 5]-code), using the primitive cyclic code C(A) with length 63 = 43−1, defining set A = {0,1,2,47}, and minimum distance d ≥ |{−1,0,1,2}|+1 = 5 (BCH-bound) [i]
- linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using the primitive cyclic code C(A) with length 63 = 43−1, defining set A = {0,1,2,3,47}, and minimum distance d ≥ |{−1,0,…,4}|+1 = 7 (BCH-bound) [i]
- linear OA(47, 63, F4, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,4)), using the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(40, 3, F4, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) for arbitrarily large s, using
- linear OA(41, 4, F4, 1) (dual of [4, 3, 2]-code), using
- Reed–Solomon code RS(3,4) [i]
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(414, 65, F4, 2, 6) (dual of [(65, 2), 116, 7]-NRT-code) | [i] | Embedding of OOA with Gilbert–Varšamov Bound | |
| 2 | Linear OOA(414, 65, F4, 3, 6) (dual of [(65, 3), 181, 7]-NRT-code) | [i] | ||
| 3 | Digital (8, 14, 65)-net over F4 | [i] | ||