Best Known (218, 218+19, s)-Nets in Base 2
(218, 218+19, 932115)-Net over F2 — Constructive and digital
Digital (218, 237, 932115)-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, 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 (20, 29, 49)-net over F2, using
(218, 218+19, 1398160)-Net over F2 — Digital
Digital (218, 237, 1398160)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2237, 1398160, F2, 6, 19) (dual of [(1398160, 6), 8388723, 20]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(229, 60, F2, 6, 9) (dual of [(60, 6), 331, 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(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(229, 60, F2, 6, 9) (dual of [(60, 6), 331, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
(218, 218+19, large)-Net in Base 2 — Upper bound on s
There is no (218, 237, large)-net in base 2, because
- 17 times m-reduction [i] would yield (218, 220, large)-net in base 2, but