Best Known (197, 197+17, s)-Nets in Base 2
(197, 197+17, 1048635)-Net over F2 — Constructive and digital
Digital (197, 214, 1048635)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (21, 29, 60)-net over F2, using
- net defined by OOA [i] based on linear OOA(229, 60, F2, 8, 8) (dual of [(60, 8), 451, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(229, 60, F2, 7, 8) (dual of [(60, 7), 391, 9]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(210, 32, F2, 7, 4) (dual of [(32, 7), 214, 5]-NRT-code), using
- appending 3 arbitrary columns [i] based on linear OOA(210, 32, F2, 4, 4) (dual of [(32, 4), 118, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(210, 32, F2, 3, 4) (dual of [(32, 3), 86, 5]-NRT-code), using
- extracting embedded OOA [i] based on digital (6, 10, 32)-net over F2, using
- appending kth column [i] based on linear OOA(210, 32, F2, 3, 4) (dual of [(32, 3), 86, 5]-NRT-code), using
- appending 3 arbitrary columns [i] based on linear OOA(210, 32, F2, 4, 4) (dual of [(32, 4), 118, 5]-NRT-code), using
- linear OOA(219, 30, F2, 7, 8) (dual of [(30, 7), 191, 9]-NRT-code), using
- extracting embedded OOA [i] based on digital (11, 19, 30)-net over F2, using
- linear OOA(210, 32, F2, 7, 4) (dual of [(32, 7), 214, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(229, 60, F2, 7, 8) (dual of [(60, 7), 391, 9]-NRT-code), using
- net defined by OOA [i] based on linear OOA(229, 60, F2, 8, 8) (dual of [(60, 8), 451, 9]-NRT-code), using
- digital (168, 185, 1048575)-net over F2, using
- net defined by OOA [i] based on linear OOA(2185, 1048575, F2, 17, 17) (dual of [(1048575, 17), 17825590, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2185, 8388601, F2, 17) (dual of [8388601, 8388416, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2185, large, F2, 17) (dual of [large, large−185, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 8388609 | 246−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2185, large, F2, 17) (dual of [large, large−185, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2185, 8388601, F2, 17) (dual of [8388601, 8388416, 18]-code), using
- net defined by OOA [i] based on linear OOA(2185, 1048575, F2, 17, 17) (dual of [(1048575, 17), 17825590, 18]-NRT-code), using
- digital (21, 29, 60)-net over F2, using
(197, 197+17, 1677789)-Net over F2 — Digital
Digital (197, 214, 1677789)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2214, 1677789, F2, 5, 17) (dual of [(1677789, 5), 8388731, 18]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(229, 69, F2, 5, 8) (dual of [(69, 5), 316, 9]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(229, 69, F2, 8) (dual of [69, 40, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(229, 135, F2, 8) (dual of [135, 106, 9]-code), using
- 1 times truncation [i] based on linear OA(230, 136, F2, 9) (dual of [136, 106, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(229, 128, F2, 9) (dual of [128, 99, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 127 = 27−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(222, 128, F2, 7) (dual of [128, 106, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 127 = 27−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(21, 8, F2, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- 1 times truncation [i] based on linear OA(230, 136, F2, 9) (dual of [136, 106, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(229, 135, F2, 8) (dual of [135, 106, 9]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(229, 69, F2, 8) (dual of [69, 40, 9]-code), using
- linear OOA(2185, 1677720, F2, 5, 17) (dual of [(1677720, 5), 8388415, 18]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2185, 8388600, F2, 17) (dual of [8388600, 8388415, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2185, large, F2, 17) (dual of [large, large−185, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 8388609 | 246−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2185, large, F2, 17) (dual of [large, large−185, 18]-code), using
- OOA 5-folding [i] based on linear OA(2185, 8388600, F2, 17) (dual of [8388600, 8388415, 18]-code), using
- linear OOA(229, 69, F2, 5, 8) (dual of [(69, 5), 316, 9]-NRT-code), using
- (u, u+v)-construction [i] based on
(197, 197+17, large)-Net in Base 2 — Upper bound on s
There is no (197, 214, large)-net in base 2, because
- 15 times m-reduction [i] would yield (197, 199, large)-net in base 2, but