Best Known (214, 214+19, s)-Nets in Base 2
(214, 214+19, 932100)-Net over F2 — Constructive and digital
Digital (214, 233, 932100)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (16, 25, 34)-net over F2, using
- net defined by OOA [i] based on linear OOA(225, 34, F2, 9, 9) (dual of [(34, 9), 281, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(225, 34, F2, 8, 9) (dual of [(34, 8), 247, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(28, 17, F2, 8, 4) (dual of [(17, 8), 128, 5]-NRT-code), using
- appending 4 arbitrary columns [i] based on linear OOA(28, 17, F2, 4, 4) (dual of [(17, 4), 60, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(28, 17, F2, 3, 4) (dual of [(17, 3), 43, 5]-NRT-code), using
- extracting embedded OOA [i] based on digital (4, 8, 17)-net over F2, using
- appending kth column [i] based on linear OOA(28, 17, F2, 3, 4) (dual of [(17, 3), 43, 5]-NRT-code), using
- appending 4 arbitrary columns [i] based on linear OOA(28, 17, F2, 4, 4) (dual of [(17, 4), 60, 5]-NRT-code), using
- linear OOA(217, 17, F2, 8, 9) (dual of [(17, 8), 119, 10]-NRT-code), using
- extracting embedded OOA [i] based on digital (8, 17, 17)-net over F2, using
- linear OOA(28, 17, F2, 8, 4) (dual of [(17, 8), 128, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(225, 34, F2, 8, 9) (dual of [(34, 8), 247, 10]-NRT-code), using
- net defined by OOA [i] based on linear OOA(225, 34, F2, 9, 9) (dual of [(34, 9), 281, 10]-NRT-code), using
- digital (189, 208, 932066)-net over F2, using
- net defined by OOA [i] based on linear OOA(2208, 932066, F2, 19, 19) (dual of [(932066, 19), 17709046, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2208, 8388595, F2, 19) (dual of [8388595, 8388387, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2208, large, F2, 19) (dual of [large, large−208, 20]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(2208, large, F2, 19) (dual of [large, large−208, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2208, 8388595, F2, 19) (dual of [8388595, 8388387, 20]-code), using
- net defined by OOA [i] based on linear OOA(2208, 932066, F2, 19, 19) (dual of [(932066, 19), 17709046, 20]-NRT-code), using
- digital (16, 25, 34)-net over F2, using
(214, 214+19, 1398134)-Net over F2 — Digital
Digital (214, 233, 1398134)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2233, 1398134, F2, 6, 19) (dual of [(1398134, 6), 8388571, 20]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(225, 34, F2, 6, 9) (dual of [(34, 6), 179, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(28, 17, F2, 6, 4) (dual of [(17, 6), 94, 5]-NRT-code), using
- appending 2 arbitrary columns [i] based on linear OOA(28, 17, F2, 4, 4) (dual of [(17, 4), 60, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(28, 17, F2, 3, 4) (dual of [(17, 3), 43, 5]-NRT-code), using
- extracting embedded OOA [i] based on digital (4, 8, 17)-net over F2, using
- appending kth column [i] based on linear OOA(28, 17, F2, 3, 4) (dual of [(17, 3), 43, 5]-NRT-code), using
- appending 2 arbitrary columns [i] based on linear OOA(28, 17, F2, 4, 4) (dual of [(17, 4), 60, 5]-NRT-code), using
- linear OOA(217, 17, F2, 6, 9) (dual of [(17, 6), 85, 10]-NRT-code), using
- extracting embedded OOA [i] based on digital (8, 17, 17)-net over F2, using
- linear OOA(28, 17, F2, 6, 4) (dual of [(17, 6), 94, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(2208, 1398100, F2, 6, 19) (dual of [(1398100, 6), 8388392, 20]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2208, 8388600, F2, 19) (dual of [8388600, 8388392, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2208, large, F2, 19) (dual of [large, large−208, 20]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(2208, large, F2, 19) (dual of [large, large−208, 20]-code), using
- OOA 6-folding [i] based on linear OA(2208, 8388600, F2, 19) (dual of [8388600, 8388392, 20]-code), using
- linear OOA(225, 34, F2, 6, 9) (dual of [(34, 6), 179, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
(214, 214+19, large)-Net in Base 2 — Upper bound on s
There is no (214, 233, large)-net in base 2, because
- 17 times m-reduction [i] would yield (214, 216, large)-net in base 2, but