Best Known (219, 238, s)-Nets in Base 2
(219, 238, 932118)-Net over F2 — Constructive and digital
Digital (219, 238, 932118)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (21, 30, 52)-net over F2, using
- net defined by OOA [i] based on linear OOA(230, 52, F2, 9, 9) (dual of [(52, 9), 438, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(230, 52, F2, 8, 9) (dual of [(52, 8), 386, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(210, 32, F2, 8, 4) (dual of [(32, 8), 246, 5]-NRT-code), using
- appending 4 arbitrary columns [i] based on linear OOA(210, 32, F2, 4, 4) (dual of [(32, 4), 118, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(210, 32, F2, 3, 4) (dual of [(32, 3), 86, 5]-NRT-code), using
- extracting embedded OOA [i] based on digital (6, 10, 32)-net over F2, using
- appending kth column [i] based on linear OOA(210, 32, F2, 3, 4) (dual of [(32, 3), 86, 5]-NRT-code), using
- appending 4 arbitrary columns [i] based on linear OOA(210, 32, F2, 4, 4) (dual of [(32, 4), 118, 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(210, 32, F2, 8, 4) (dual of [(32, 8), 246, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(230, 52, F2, 8, 9) (dual of [(52, 8), 386, 10]-NRT-code), using
- net defined by OOA [i] based on linear OOA(230, 52, F2, 9, 9) (dual of [(52, 9), 438, 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 (21, 30, 52)-net over F2, using
(219, 238, 1408589)-Net over F2 — Digital
Digital (219, 238, 1408589)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2238, 1408589, F2, 5, 19) (dual of [(1408589, 5), 7042707, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2238, 1677772, F2, 5, 19) (dual of [(1677772, 5), 8388622, 20]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(230, 52, F2, 5, 9) (dual of [(52, 5), 230, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(210, 32, F2, 5, 4) (dual of [(32, 5), 150, 5]-NRT-code), using
- appending 1 arbitrary column [i] based on linear OOA(210, 32, F2, 4, 4) (dual of [(32, 4), 118, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(210, 32, F2, 3, 4) (dual of [(32, 3), 86, 5]-NRT-code), using
- extracting embedded OOA [i] based on digital (6, 10, 32)-net over F2, using
- appending kth column [i] based on linear OOA(210, 32, F2, 3, 4) (dual of [(32, 3), 86, 5]-NRT-code), using
- appending 1 arbitrary column [i] based on linear OOA(210, 32, F2, 4, 4) (dual of [(32, 4), 118, 5]-NRT-code), using
- linear OOA(220, 26, F2, 5, 9) (dual of [(26, 5), 110, 10]-NRT-code), using
- extracting embedded OOA [i] based on digital (11, 20, 26)-net over F2, using
- linear OOA(210, 32, F2, 5, 4) (dual of [(32, 5), 150, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(2208, 1677720, F2, 5, 19) (dual of [(1677720, 5), 8388392, 20]-NRT-code), using
- OOA 5-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 5-folding [i] based on linear OA(2208, 8388600, F2, 19) (dual of [8388600, 8388392, 20]-code), using
- linear OOA(230, 52, F2, 5, 9) (dual of [(52, 5), 230, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OOA(2238, 1677772, F2, 5, 19) (dual of [(1677772, 5), 8388622, 20]-NRT-code), using
(219, 238, large)-Net in Base 2 — Upper bound on s
There is no (219, 238, large)-net in base 2, because
- 17 times m-reduction [i] would yield (219, 221, large)-net in base 2, but