Information on Result #1653576
Linear OOA(2212, 1677774, F2, 5, 17) (dual of [(1677774, 5), 8388658, 18]-NRT-code), using (u, u+v)-construction based on
- linear OOA(227, 54, F2, 5, 8) (dual of [(54, 5), 243, 9]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(227, 54, F2, 8) (dual of [54, 27, 9]-code), using
- discarding factors / shortening the dual code based on 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}) [i] 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)
- 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}) [i] based on
- discarding factors / shortening the dual code based on linear OA(227, 78, F2, 8) (dual of [78, 51, 9]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(227, 54, F2, 8) (dual of [54, 27, 9]-code), using
- linear OOA(2185, 1677720, F2, 5, 17) (dual of [(1677720, 5), 8388415, 18]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2185, 8388600, F2, 17) (dual of [8388600, 8388415, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2185, large, F2, 17) (dual of [large, large−185, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 8388609 | 246−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2185, large, F2, 17) (dual of [large, large−185, 18]-code), using
- OOA 5-folding [i] based on linear OA(2185, 8388600, F2, 17) (dual of [8388600, 8388415, 18]-code), using
Mode: Linear.
Optimality
Show details for fixed k and m, k and s, k and t, m and s, 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(2212, 1677774, F2, 6, 17) (dual of [(1677774, 6), 10066432, 18]-NRT-code) | [i] | Embedding of OOA with Gilbert–Varšamov Bound | |
| 2 | Linear OOA(2212, 1677774, F2, 7, 17) (dual of [(1677774, 7), 11744206, 18]-NRT-code) | [i] | ||
| 3 | Linear OOA(2212, 1677774, F2, 8, 17) (dual of [(1677774, 8), 13421980, 18]-NRT-code) | [i] | ||
| 4 | Digital (195, 212, 1677774)-net over F2 | [i] | ||