Best Known (57−4, 57, s)-Nets in Base 7
(57−4, 57, large)-Net over F7 — Constructive and digital
Digital (53, 57, large)-net over F7, using
- 713 times duplication [i] based on digital (40, 44, large)-net over F7, using
- t-expansion [i] based on digital (39, 44, large)-net over F7, using
- trace code for nets [i] based on digital (17, 22, 5764803)-net over F49, using
- net defined by OOA [i] based on linear OOA(4922, 5764803, F49, 6, 5) (dual of [(5764803, 6), 34588796, 6]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(4922, 5764804, F49, 2, 5) (dual of [(5764804, 2), 11529586, 6]-NRT-code), using
- (u, u+v)-construction [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
- 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(495, 5884901, F49, 2, 2) (dual of [(5884901, 2), 11769797, 3]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(4922, 5764804, F49, 2, 5) (dual of [(5764804, 2), 11529586, 6]-NRT-code), using
- net defined by OOA [i] based on linear OOA(4922, 5764803, F49, 6, 5) (dual of [(5764803, 6), 34588796, 6]-NRT-code), using
- trace code for nets [i] based on digital (17, 22, 5764803)-net over F49, using
- t-expansion [i] based on digital (39, 44, large)-net over F7, using
(57−4, 57, large)-Net in Base 7 — Upper bound on s
There is no (53, 57, large)-net in base 7, because
- 2 times m-reduction [i] would yield (53, 55, large)-net in base 7, but