Best Known (218, 218+18, s)-Nets in Base 2
(218, 218+18, 932116)-Net over F2 — Constructive and digital
Digital (218, 236, 932116)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (20, 29, 49)-net over F2, using
- net defined by OOA [i] based on linear OOA(229, 49, F2, 9, 9) (dual of [(49, 9), 412, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(229, 49, F2, 8, 9) (dual of [(49, 8), 363, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(29, 23, F2, 8, 4) (dual of [(23, 8), 175, 5]-NRT-code), using
- appending 4 arbitrary columns [i] based on linear OOA(29, 23, F2, 4, 4) (dual of [(23, 4), 83, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(29, 23, F2, 3, 4) (dual of [(23, 3), 60, 5]-NRT-code), using
- extracting embedded OOA [i] based on digital (5, 9, 23)-net over F2, using
- appending kth column [i] based on linear OOA(29, 23, F2, 3, 4) (dual of [(23, 3), 60, 5]-NRT-code), using
- appending 4 arbitrary columns [i] based on linear OOA(29, 23, F2, 4, 4) (dual of [(23, 4), 83, 5]-NRT-code), using
- linear OOA(220, 26, F2, 8, 9) (dual of [(26, 8), 188, 10]-NRT-code), using
- extracting embedded OOA [i] based on digital (11, 20, 26)-net over F2, using
- linear OOA(29, 23, F2, 8, 4) (dual of [(23, 8), 175, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(229, 49, F2, 8, 9) (dual of [(49, 8), 363, 10]-NRT-code), using
- net defined by OOA [i] based on linear OOA(229, 49, F2, 9, 9) (dual of [(49, 9), 412, 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 (20, 29, 49)-net over F2, using
(218, 218+18, 1677780)-Net over F2 — Digital
Digital (218, 236, 1677780)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2236, 1677780, F2, 5, 18) (dual of [(1677780, 5), 8388664, 19]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(229, 60, F2, 5, 9) (dual of [(60, 5), 271, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(229, 60, F2, 2, 9) (dual of [(60, 2), 91, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(229, 64, F2, 2, 9) (dual of [(64, 2), 99, 10]-NRT-code), using
- OOA 2-folding [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]
- OOA 2-folding [i] based on linear OA(229, 128, F2, 9) (dual of [128, 99, 10]-code), using
- discarding factors / shortening the dual code based on linear OOA(229, 64, F2, 2, 9) (dual of [(64, 2), 99, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(229, 60, F2, 2, 9) (dual of [(60, 2), 91, 10]-NRT-code), using
- 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(229, 60, F2, 5, 9) (dual of [(60, 5), 271, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
(218, 218+18, large)-Net in Base 2 — Upper bound on s
There is no (218, 236, large)-net in base 2, because
- 16 times m-reduction [i] would yield (218, 220, large)-net in base 2, but