Information on Result #1616127
Linear OOA(2251, 2097525, F2, 4, 18) (dual of [(2097525, 4), 8389849, 19]-NRT-code), using (u, u+v)-construction based on
- linear OOA(244, 375, F2, 4, 9) (dual of [(375, 4), 1456, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(244, 375, F2, 2, 9) (dual of [(375, 2), 706, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(244, 523, F2, 2, 9) (dual of [(523, 2), 1002, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(244, 1046, F2, 9) (dual of [1046, 1002, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(244, 1047, F2, 9) (dual of [1047, 1003, 10]-code), using
- adding a parity check bit [i] based on linear OA(243, 1046, F2, 8) (dual of [1046, 1003, 9]-code), using
- construction XX applied to C1 = C([1021,4]), C2 = C([1,6]), C3 = C1 + C2 = C([1,4]), and C∩ = C1 ∩ C2 = C([1021,6]) [i] based on
- linear OA(231, 1023, F2, 7) (dual of [1023, 992, 8]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,4}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(230, 1023, F2, 6) (dual of [1023, 993, 7]-code), using the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(241, 1023, F2, 9) (dual of [1023, 982, 10]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,6}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(220, 1023, F2, 4) (dual of [1023, 1003, 5]-code), using the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(21, 12, F2, 1) (dual of [12, 11, 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, 11, F2, 1) (dual of [11, 10, 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([1021,4]), C2 = C([1,6]), C3 = C1 + C2 = C([1,4]), and C∩ = C1 ∩ C2 = C([1021,6]) [i] based on
- adding a parity check bit [i] based on linear OA(243, 1046, F2, 8) (dual of [1046, 1003, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(244, 1047, F2, 9) (dual of [1047, 1003, 10]-code), using
- OOA 2-folding [i] based on linear OA(244, 1046, F2, 9) (dual of [1046, 1002, 10]-code), using
- discarding factors / shortening the dual code based on linear OOA(244, 523, F2, 2, 9) (dual of [(523, 2), 1002, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(244, 375, F2, 2, 9) (dual of [(375, 2), 706, 10]-NRT-code), using
- linear OOA(2207, 2097150, F2, 4, 18) (dual of [(2097150, 4), 8388393, 19]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2207, 8388600, F2, 18) (dual of [8388600, 8388393, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(2207, large, F2, 18) (dual of [large, large−207, 19]-code), using
- the primitive narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- discarding factors / shortening the dual code based on linear OA(2207, large, F2, 18) (dual of [large, large−207, 19]-code), using
- OOA 4-folding [i] based on linear OA(2207, 8388600, F2, 18) (dual of [8388600, 8388393, 19]-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(2251, 2097525, F2, 5, 18) (dual of [(2097525, 5), 10487374, 19]-NRT-code) | [i] | Embedding of OOA with Gilbert–Varšamov Bound | |
| 2 | Linear OOA(2251, 2097525, F2, 6, 18) (dual of [(2097525, 6), 12584899, 19]-NRT-code) | [i] | ||
| 3 | Linear OOA(2251, 2097525, F2, 7, 18) (dual of [(2097525, 7), 14682424, 19]-NRT-code) | [i] | ||
| 4 | Linear OOA(2251, 2097525, F2, 8, 18) (dual of [(2097525, 8), 16779949, 19]-NRT-code) | [i] | ||
| 5 | Digital (233, 251, 2097525)-net over F2 | [i] | ||