Best Known (25−5, 25, s)-Nets in Base 49
(25−5, 25, large)-Net over F49 — Constructive and digital
Digital (20, 25, large)-net over F49, using
- (u, u+v)-construction [i] based on
- digital (0, 2, 50)-net over F49, using
- digital (18, 23, 8388602)-net over F49, using
- net defined by OOA [i] based on linear OOA(4923, 8388602, F49, 6, 5) (dual of [(8388602, 6), 50331589, 6]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(4923, large, F49, 2, 5), using
- generalized (u, u+v)-construction [i] based on
- linear OOA(491, 2882402, F49, 2, 1) (dual of [(2882402, 2), 5764803, 2]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(491, s, F49, 2, 1) with arbitrarily large s, using
- appending 1 arbitrary column [i] based on linear OA(491, s, F49, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- discarding factors / shortening the dual code based on linear OOA(491, s, F49, 2, 1) with arbitrarily large s, using
- linear OOA(495, 2882402, F49, 2, 2) (dual of [(2882402, 2), 5764799, 3]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(495, 5884901, F49, 2, 2) (dual of [(5884901, 2), 11769797, 3]-NRT-code), using
- appending kth column [i] based on linear OA(495, 5884901, F49, 2) (dual of [5884901, 5884896, 3]-code), using
- Hamming code H(5,49) [i]
- appending kth column [i] based on linear OA(495, 5884901, F49, 2) (dual of [5884901, 5884896, 3]-code), using
- discarding factors / shortening the dual code based on linear OOA(495, 5884901, F49, 2, 2) (dual of [(5884901, 2), 11769797, 3]-NRT-code), using
- linear OOA(4917, 2882402, F49, 2, 5) (dual of [(2882402, 2), 5764787, 6]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4917, 5764804, F49, 5) (dual of [5764804, 5764787, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(4917, 5764805, F49, 5) (dual of [5764805, 5764788, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(4917, 5764801, F49, 5) (dual of [5764801, 5764784, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(4913, 5764801, F49, 4) (dual of [5764801, 5764788, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(490, 4, F49, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(490, s, F49, 0) (dual of [s, s, 1]-code) with 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(4917, 5764805, F49, 5) (dual of [5764805, 5764788, 6]-code), using
- OOA 2-folding [i] based on linear OA(4917, 5764804, F49, 5) (dual of [5764804, 5764787, 6]-code), using
- linear OOA(491, 2882402, F49, 2, 1) (dual of [(2882402, 2), 5764803, 2]-NRT-code), using
- generalized (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(4923, large, F49, 2, 5), using
- net defined by OOA [i] based on linear OOA(4923, 8388602, F49, 6, 5) (dual of [(8388602, 6), 50331589, 6]-NRT-code), using
(25−5, 25, large)-Net in Base 49 — Upper bound on s
There is no (20, 25, large)-net in base 49, because
- 3 times m-reduction [i] would yield (20, 22, large)-net in base 49, but