Best Known (219, 237, s)-Nets in Base 2
(219, 237, 932119)-Net over F2 — Constructive and digital
Digital (219, 237, 932119)-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, 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 (21, 30, 52)-net over F2, using
(219, 237, 1677788)-Net over F2 — Digital
Digital (219, 237, 1677788)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2237, 1677788, F2, 5, 18) (dual of [(1677788, 5), 8388703, 19]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(230, 68, F2, 5, 9) (dual of [(68, 5), 310, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(230, 68, F2, 2, 9) (dual of [(68, 2), 106, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(230, 136, F2, 9) (dual of [136, 106, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(6) [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]
- linear OA(222, 128, F2, 7) (dual of [128, 106, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 127 = 27−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(21, 8, F2, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- OOA 2-folding [i] based on linear OA(230, 136, F2, 9) (dual of [136, 106, 10]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(230, 68, F2, 2, 9) (dual of [(68, 2), 106, 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(230, 68, F2, 5, 9) (dual of [(68, 5), 310, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
(219, 237, large)-Net in Base 2 — Upper bound on s
There is no (219, 237, large)-net in base 2, because
- 16 times m-reduction [i] would yield (219, 221, large)-net in base 2, but