Best Known (232−18, 232, s)-Nets in Base 2
(232−18, 232, 932101)-Net over F2 — Constructive and digital
Digital (214, 232, 932101)-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, 207, 932067)-net over F2, using
- net defined by OOA [i] based on linear OOA(2207, 932067, F2, 18, 18) (dual of [(932067, 18), 16776999, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(2207, large, F2, 18) (dual of [large, large−207, 19]-code), using
- the primitive narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- OA 9-folding and stacking [i] based on linear OA(2207, large, F2, 18) (dual of [large, large−207, 19]-code), using
- net defined by OOA [i] based on linear OOA(2207, 932067, F2, 18, 18) (dual of [(932067, 18), 16776999, 19]-NRT-code), using
- digital (16, 25, 34)-net over F2, using
(232−18, 232, 1677754)-Net over F2 — Digital
Digital (214, 232, 1677754)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2232, 1677754, F2, 5, 18) (dual of [(1677754, 5), 8388538, 19]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(225, 34, F2, 5, 9) (dual of [(34, 5), 145, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(28, 17, F2, 5, 4) (dual of [(17, 5), 77, 5]-NRT-code), using
- appending 1 arbitrary column [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 1 arbitrary column [i] based on linear OOA(28, 17, F2, 4, 4) (dual of [(17, 4), 60, 5]-NRT-code), using
- linear OOA(217, 17, F2, 5, 9) (dual of [(17, 5), 68, 10]-NRT-code), using
- extracting embedded OOA [i] based on digital (8, 17, 17)-net over F2, using
- linear OOA(28, 17, F2, 5, 4) (dual of [(17, 5), 77, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(2207, 1677720, F2, 5, 18) (dual of [(1677720, 5), 8388393, 19]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2207, 8388600, F2, 18) (dual of [8388600, 8388393, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(2207, large, F2, 18) (dual of [large, large−207, 19]-code), using
- the primitive narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- discarding factors / shortening the dual code based on linear OA(2207, large, F2, 18) (dual of [large, large−207, 19]-code), using
- OOA 5-folding [i] based on linear OA(2207, 8388600, F2, 18) (dual of [8388600, 8388393, 19]-code), using
- linear OOA(225, 34, F2, 5, 9) (dual of [(34, 5), 145, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
(232−18, 232, large)-Net in Base 2 — Upper bound on s
There is no (214, 232, large)-net in base 2, because
- 16 times m-reduction [i] would yield (214, 216, large)-net in base 2, but