Best Known (218−17, 218, s)-Nets in Base 2
(218−17, 218, 1048653)-Net over F2 — Constructive and digital
Digital (201, 218, 1048653)-net over F2, using
- 21 times duplication [i] based on digital (200, 217, 1048653)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (24, 32, 78)-net over F2, using
- net defined by OOA [i] based on linear OOA(232, 78, F2, 8, 8) (dual of [(78, 8), 592, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(232, 78, F2, 7, 8) (dual of [(78, 7), 514, 9]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(211, 47, F2, 7, 4) (dual of [(47, 7), 318, 5]-NRT-code), using
- appending 3 arbitrary columns [i] based on linear OOA(211, 47, F2, 4, 4) (dual of [(47, 4), 177, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(211, 47, F2, 3, 4) (dual of [(47, 3), 130, 5]-NRT-code), using
- extracting embedded OOA [i] based on digital (7, 11, 47)-net over F2, using
- appending kth column [i] based on linear OOA(211, 47, F2, 3, 4) (dual of [(47, 3), 130, 5]-NRT-code), using
- appending 3 arbitrary columns [i] based on linear OOA(211, 47, F2, 4, 4) (dual of [(47, 4), 177, 5]-NRT-code), using
- linear OOA(221, 39, F2, 7, 8) (dual of [(39, 7), 252, 9]-NRT-code), using
- extracting embedded OOA [i] based on digital (13, 21, 39)-net over F2, using
- linear OOA(211, 47, F2, 7, 4) (dual of [(47, 7), 318, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(232, 78, F2, 7, 8) (dual of [(78, 7), 514, 9]-NRT-code), using
- net defined by OOA [i] based on linear OOA(232, 78, F2, 8, 8) (dual of [(78, 8), 592, 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 (24, 32, 78)-net over F2, using
- (u, u+v)-construction [i] based on
(218−17, 218, 1677852)-Net over F2 — Digital
Digital (201, 218, 1677852)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2218, 1677852, F2, 5, 17) (dual of [(1677852, 5), 8389042, 18]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(233, 132, F2, 5, 8) (dual of [(132, 5), 627, 9]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(233, 132, F2, 2, 8) (dual of [(132, 2), 231, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(233, 264, F2, 8) (dual of [264, 231, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(233, 265, F2, 8) (dual of [265, 232, 9]-code), using
- 1 times truncation [i] based on linear OA(234, 266, F2, 9) (dual of [266, 232, 10]-code), using
- construction X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(233, 256, F2, 9) (dual of [256, 223, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(225, 256, F2, 7) (dual of [256, 231, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(29, 10, F2, 9) (dual of [10, 1, 10]-code or 10-arc in PG(8,2)), using
- dual of repetition code with length 10 [i]
- linear OA(21, 10, F2, 1) (dual of [10, 9, 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 X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- 1 times truncation [i] based on linear OA(234, 266, F2, 9) (dual of [266, 232, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(233, 265, F2, 8) (dual of [265, 232, 9]-code), using
- OOA 2-folding [i] based on linear OA(233, 264, F2, 8) (dual of [264, 231, 9]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(233, 132, F2, 2, 8) (dual of [(132, 2), 231, 9]-NRT-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(233, 132, F2, 5, 8) (dual of [(132, 5), 627, 9]-NRT-code), using
- (u, u+v)-construction [i] based on
(218−17, 218, large)-Net in Base 2 — Upper bound on s
There is no (201, 218, large)-net in base 2, because
- 15 times m-reduction [i] would yield (201, 203, large)-net in base 2, but