Best Known (159−12, 159, s)-Nets in Base 2
(159−12, 159, 1398227)-Net over F2 — Constructive and digital
Digital (147, 159, 1398227)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (15, 21, 127)-net over F2, using
- digital (126, 138, 1398100)-net over F2, using
- net defined by OOA [i] based on linear OOA(2138, 1398100, F2, 12, 12) (dual of [(1398100, 12), 16777062, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(2138, 8388600, F2, 12) (dual of [8388600, 8388462, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(2138, large, F2, 12) (dual of [large, large−138, 13]-code), using
- the primitive narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(2138, large, F2, 12) (dual of [large, large−138, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(2138, 8388600, F2, 12) (dual of [8388600, 8388462, 13]-code), using
- net defined by OOA [i] based on linear OOA(2138, 1398100, F2, 12, 12) (dual of [(1398100, 12), 16777062, 13]-NRT-code), using
(159−12, 159, 2791086)-Net over F2 — Digital
Digital (147, 159, 2791086)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2159, 2791086, F2, 3, 12) (dual of [(2791086, 3), 8373099, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2159, 2796328, F2, 3, 12) (dual of [(2796328, 3), 8388825, 13]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(221, 127, F2, 3, 6) (dual of [(127, 3), 360, 7]-NRT-code), using
- extracting embedded OOA [i] based on digital (15, 21, 127)-net over F2, using
- linear OOA(2138, 2796201, F2, 3, 12) (dual of [(2796201, 3), 8388465, 13]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2138, large, F2, 12) (dual of [large, large−138, 13]-code), using
- the primitive narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- OOA 3-folding [i] based on linear OA(2138, large, F2, 12) (dual of [large, large−138, 13]-code), using
- linear OOA(221, 127, F2, 3, 6) (dual of [(127, 3), 360, 7]-NRT-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OOA(2159, 2796328, F2, 3, 12) (dual of [(2796328, 3), 8388825, 13]-NRT-code), using
(159−12, 159, large)-Net in Base 2 — Upper bound on s
There is no (147, 159, large)-net in base 2, because
- 10 times m-reduction [i] would yield (147, 149, large)-net in base 2, but