Information on Result #3178418
Digital (72, 77, large)-net over F81, using 5 times m-reduction based on digital (72, 82, large)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (12, 17, 4194301)-net over F81, using
- net defined by OOA [i] based on linear OOA(8117, 4194301, F81, 5, 5) (dual of [(4194301, 5), 20971488, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(8117, 4194301, F81, 4, 5) (dual of [(4194301, 4), 16777187, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(8117, large, F81, 5) (dual of [large, large−17, 6]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- OOA 2-folding and stacking with additional row [i] based on linear OA(8117, large, F81, 5) (dual of [large, large−17, 6]-code), using
- appending kth column [i] based on linear OOA(8117, 4194301, F81, 4, 5) (dual of [(4194301, 4), 16777187, 6]-NRT-code), using
- net defined by OOA [i] based on linear OOA(8117, 4194301, F81, 5, 5) (dual of [(4194301, 5), 20971488, 6]-NRT-code), using
- digital (55, 65, 4194383)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (0, 5, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- digital (50, 60, 4194301)-net over F81, using
- net defined by OOA [i] based on linear OOA(8160, 4194301, F81, 15, 10) (dual of [(4194301, 15), 62914455, 11]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(8160, large, F81, 3, 10), using
- generalized (u, u+v)-construction [i] based on
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(810, s, F81, 3, 0) for arbitrarily large s, using
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
- linear OOA(811, 177148, F81, 3, 1) (dual of [(177148, 3), 531443, 2]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(811, s, F81, 3, 1) for arbitrarily large s, using
- appending 2 arbitrary columns [i] based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) for arbitrarily large s, using
- discarding factors / shortening the dual code based on linear OOA(811, s, F81, 3, 1) for arbitrarily large s, using
- linear OOA(811, 177148, F81, 3, 1) (dual of [(177148, 3), 531443, 2]-NRT-code) (see above)
- linear OOA(811, 177148, F81, 3, 1) (dual of [(177148, 3), 531443, 2]-NRT-code) (see above)
- linear OOA(811, 177148, F81, 3, 1) (dual of [(177148, 3), 531443, 2]-NRT-code) (see above)
- linear OOA(811, 177148, F81, 3, 1) (dual of [(177148, 3), 531443, 2]-NRT-code) (see above)
- linear OOA(814, 177148, F81, 3, 2) (dual of [(177148, 3), 531440, 3]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(814, 538084, F81, 3, 2) (dual of [(538084, 3), 1614248, 3]-NRT-code), using
- appending 1 arbitrary column [i] based on linear OOA(814, 538084, F81, 2, 2) (dual of [(538084, 2), 1076164, 3]-NRT-code), using
- appending kth column [i] based on linear OA(814, 538084, F81, 2) (dual of [538084, 538080, 3]-code), using
- Hamming code H(4,81) [i]
- appending kth column [i] based on linear OA(814, 538084, F81, 2) (dual of [538084, 538080, 3]-code), using
- appending 1 arbitrary column [i] based on linear OOA(814, 538084, F81, 2, 2) (dual of [(538084, 2), 1076164, 3]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(814, 538084, F81, 3, 2) (dual of [(538084, 3), 1614248, 3]-NRT-code), using
- linear OOA(814, 177148, F81, 3, 2) (dual of [(177148, 3), 531440, 3]-NRT-code) (see above)
- linear OOA(816, 177148, F81, 3, 3) (dual of [(177148, 3), 531438, 4]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(816, 538247, F81, 3, 3) (dual of [(538247, 3), 1614735, 4]-NRT-code), using
- linear OOA(8113, 177148, F81, 3, 5) (dual of [(177148, 3), 531431, 6]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8113, 265721, F81, 3, 5) (dual of [(265721, 3), 797150, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(8113, 531443, F81, 5) (dual of [531443, 531430, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(8113, 531444, F81, 5) (dual of [531444, 531431, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(8113, 531441, F81, 5) (dual of [531441, 531428, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(8110, 531441, F81, 4) (dual of [531441, 531431, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(810, 3, F81, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(810, s, F81, 0) (dual of [s, s, 1]-code) for arbitrarily large s, using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- discarding factors / shortening the dual code based on linear OA(8113, 531444, F81, 5) (dual of [531444, 531431, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(8113, 531443, F81, 5) (dual of [531443, 531430, 6]-code), using
- discarding factors / shortening the dual code based on linear OOA(8113, 265721, F81, 3, 5) (dual of [(265721, 3), 797150, 6]-NRT-code), using
- linear OOA(8128, 177148, F81, 3, 10) (dual of [(177148, 3), 531416, 11]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8128, 531444, F81, 10) (dual of [531444, 531416, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(8128, 531441, F81, 10) (dual of [531441, 531413, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(8125, 531441, F81, 9) (dual of [531441, 531416, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(810, 3, F81, 0) (dual of [3, 3, 1]-code) (see above)
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- OOA 3-folding [i] based on linear OA(8128, 531444, F81, 10) (dual of [531444, 531416, 11]-code), using
- linear OOA(810, 177148, F81, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code), using
- generalized (u, u+v)-construction [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OOA(8160, large, F81, 3, 10), using
- net defined by OOA [i] based on linear OOA(8160, 4194301, F81, 15, 10) (dual of [(4194301, 15), 62914455, 11]-NRT-code), using
- digital (0, 5, 82)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (12, 17, 4194301)-net over F81, using
Mode: Constructive and linear.
Optimality
Show details for fixed k and m, k and t, m and t.
Other Results with Identical Parameters
None.
Depending Results
The following results depend on this result:
| Result | This result only | Method | ||
|---|---|---|---|---|
| 1 | Digital (73, 77, large)-net over F81 | [i] | t-Expansion | |
| 2 | Digital (74, 77, large)-net over F81 | [i] | ||