Best Known (53−6, 53, s)-Nets in Base 81
(53−6, 53, large)-Net over F81 — Constructive and digital
Digital (47, 53, large)-net over F81, using
- t-expansion [i] based on digital (45, 53, large)-net over F81, using
- 1 times m-reduction [i] based on digital (45, 54, large)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 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 (41, 50, 8388602)-net over F81, using
- net defined by OOA [i] based on linear OOA(8150, 8388602, F81, 12, 9) (dual of [(8388602, 12), 100663174, 10]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(8150, large, F81, 4, 9), using
- generalized (u, u+v)-construction [i] based on
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(810, s, F81, 4, 0) with arbitrarily large s, using
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
- linear OOA(811, 132861, F81, 4, 1) (dual of [(132861, 4), 531443, 2]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(811, s, F81, 4, 1) with arbitrarily large s, using
- appending 3 arbitrary columns [i] based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- discarding factors / shortening the dual code based on linear OOA(811, s, F81, 4, 1) with arbitrarily large s, using
- linear OOA(811, 132861, F81, 4, 1) (dual of [(132861, 4), 531443, 2]-NRT-code) (see above)
- linear OOA(811, 132861, F81, 4, 1) (dual of [(132861, 4), 531443, 2]-NRT-code) (see above)
- linear OOA(811, 132861, F81, 4, 1) (dual of [(132861, 4), 531443, 2]-NRT-code) (see above)
- linear OOA(811, 132861, F81, 4, 1) (dual of [(132861, 4), 531443, 2]-NRT-code) (see above)
- linear OOA(814, 132861, F81, 4, 2) (dual of [(132861, 4), 531440, 3]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(814, 538084, F81, 4, 2) (dual of [(538084, 4), 2152332, 3]-NRT-code), using
- appending 2 arbitrary columns [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 2 arbitrary columns [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, 4, 2) (dual of [(538084, 4), 2152332, 3]-NRT-code), using
- linear OOA(816, 132861, F81, 4, 3) (dual of [(132861, 4), 531438, 4]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(816, 538248, F81, 4, 3) (dual of [(538248, 4), 2152986, 4]-NRT-code), using
- appending 1 arbitrary column [i] based on linear OOA(816, 538248, F81, 3, 3) (dual of [(538248, 3), 1614738, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(816, 538248, F81, 2, 3) (dual of [(538248, 2), 1076490, 4]-NRT-code), using
- appending 1 arbitrary column [i] based on linear OOA(816, 538248, F81, 3, 3) (dual of [(538248, 3), 1614738, 4]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(816, 538248, F81, 4, 3) (dual of [(538248, 4), 2152986, 4]-NRT-code), using
- linear OOA(8110, 132861, F81, 4, 4) (dual of [(132861, 4), 531434, 5]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8110, 265722, F81, 4, 4) (dual of [(265722, 4), 1062878, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(8110, 531444, F81, 4) (dual of [531444, 531434, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- 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(817, 531441, F81, 3) (dual of [531441, 531434, 4]-code or 531441-cap in PG(6,81)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [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) with arbitrarily large s, using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- OA 2-folding and stacking [i] based on linear OA(8110, 531444, F81, 4) (dual of [531444, 531434, 5]-code), using
- discarding factors / shortening the dual code based on linear OOA(8110, 265722, F81, 4, 4) (dual of [(265722, 4), 1062878, 5]-NRT-code), using
- linear OOA(8125, 132861, F81, 4, 9) (dual of [(132861, 4), 531419, 10]-NRT-code), using
- OOA 4-folding [i] based on linear OA(8125, 531444, F81, 9) (dual of [531444, 531419, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- 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(8122, 531441, F81, 8) (dual of [531441, 531419, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(810, 3, F81, 0) (dual of [3, 3, 1]-code) (see above)
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- OOA 4-folding [i] based on linear OA(8125, 531444, F81, 9) (dual of [531444, 531419, 10]-code), using
- linear OOA(810, 132861, F81, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code), using
- generalized (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(8150, large, F81, 4, 9), using
- net defined by OOA [i] based on linear OOA(8150, 8388602, F81, 12, 9) (dual of [(8388602, 12), 100663174, 10]-NRT-code), using
- digital (0, 4, 82)-net over F81, using
- (u, u+v)-construction [i] based on
- 1 times m-reduction [i] based on digital (45, 54, large)-net over F81, using
(53−6, 53, large)-Net in Base 81 — Upper bound on s
There is no (47, 53, large)-net in base 81, because
- 4 times m-reduction [i] would yield (47, 49, large)-net in base 81, but