Best Known (67, 67+6, s)-Nets in Base 7
(67, 67+6, large)-Net over F7 — Constructive and digital
Digital (67, 73, large)-net over F7, using
- 1 times m-reduction [i] based on digital (67, 74, large)-net over F7, using
- trace code for nets [i] based on digital (30, 37, 5764802)-net over F49, using
- net defined by OOA [i] based on linear OOA(4937, 5764802, F49, 9, 7) (dual of [(5764802, 9), 51883181, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(4937, 5764803, F49, 3, 7) (dual of [(5764803, 3), 17294372, 8]-NRT-code), using
- generalized (u, u+v)-construction [i] based on
- linear OOA(495, 1921601, F49, 3, 2) (dual of [(1921601, 3), 5764798, 3]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(495, 5884901, F49, 3, 2) (dual of [(5884901, 3), 17654698, 3]-NRT-code), using
- appending 1 arbitrary column [i] 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
- appending 1 arbitrary column [i] based on linear OOA(495, 5884901, F49, 2, 2) (dual of [(5884901, 2), 11769797, 3]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(495, 5884901, F49, 3, 2) (dual of [(5884901, 3), 17654698, 3]-NRT-code), using
- linear OOA(497, 1921601, F49, 3, 3) (dual of [(1921601, 3), 5764796, 4]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(497, 5769602, F49, 3, 3) (dual of [(5769602, 3), 17308799, 4]-NRT-code), using
- linear OOA(4925, 1921601, F49, 3, 7) (dual of [(1921601, 3), 5764778, 8]-NRT-code), using
- OOA 3-folding [i] based on linear OA(4925, 5764803, F49, 7) (dual of [5764803, 5764778, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(4925, 5764805, F49, 7) (dual of [5764805, 5764780, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(4925, 5764801, F49, 7) (dual of [5764801, 5764776, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(4921, 5764801, F49, 6) (dual of [5764801, 5764780, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(6) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(4925, 5764805, F49, 7) (dual of [5764805, 5764780, 8]-code), using
- OOA 3-folding [i] based on linear OA(4925, 5764803, F49, 7) (dual of [5764803, 5764778, 8]-code), using
- linear OOA(495, 1921601, F49, 3, 2) (dual of [(1921601, 3), 5764798, 3]-NRT-code), using
- generalized (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(4937, 5764803, F49, 3, 7) (dual of [(5764803, 3), 17294372, 8]-NRT-code), using
- net defined by OOA [i] based on linear OOA(4937, 5764802, F49, 9, 7) (dual of [(5764802, 9), 51883181, 8]-NRT-code), using
- trace code for nets [i] based on digital (30, 37, 5764802)-net over F49, using
(67, 67+6, large)-Net in Base 7 — Upper bound on s
There is no (67, 73, large)-net in base 7, because
- 4 times m-reduction [i] would yield (67, 69, large)-net in base 7, but