Information on Result #701351
Linear OA(724, 58, F7, 13) (dual of [58, 34, 14]-code), using construction XX applied to C1 = C({0,1,2,3,4,5,6,8,27,34,41}), C2 = C([0,9]), C3 = C1 + C2 = C([0,8]), and C∩ = C1 ∩ C2 = C({0,1,2,3,4,5,6,8,9,27,34,41}) based on
- linear OA(720, 48, F7, 12) (dual of [48, 28, 13]-code), using the primitive cyclic code C(A) with length 48 = 72−1, defining set A = {0,1,2,3,4,5,6,8,27,34,41}, and minimum distance d ≥ |{−3,−2,…,8}|+1 = 13 (BCH-bound) [i]
- linear OA(716, 48, F7, 10) (dual of [48, 32, 11]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 48 = 72−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(722, 48, F7, 13) (dual of [48, 26, 14]-code), using the primitive cyclic code C(A) with length 48 = 72−1, defining set A = {0,1,2,3,4,5,6,8,9,27,34,41}, and minimum distance d ≥ |{−3,−2,…,9}|+1 = 14 (BCH-bound) [i]
- linear OA(714, 48, F7, 9) (dual of [48, 34, 10]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 48 = 72−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(72, 8, F7, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,7)), using
- extended Reed–Solomon code RSe(6,7) [i]
- Hamming code H(2,7) [i]
- algebraic-geometric code AG(F, Q+1P) with degQ = 3 and degPÂ =Â 2 [i] based on function field F/F7 with g(F) = 0 and N(F) ≥ 8, using the rational function field F7(x) [i]
- linear OA(70, 2, F7, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) for arbitrarily large s, using
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 OA(725, 61, F7, 13) (dual of [61, 36, 14]-code) | [i] | Varšamov–Edel Lengthening | |
| 2 | Linear OOA(724, 29, F7, 2, 13) (dual of [(29, 2), 34, 14]-NRT-code) | [i] | OOA Folding | |