Best Known (145, 145+12, s)-Nets in Base 2
(145, 145+12, 1398172)-Net over F2 — Constructive and digital
Digital (145, 157, 1398172)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (13, 19, 72)-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
(145, 145+12, 2347013)-Net over F2 — Digital
Digital (145, 157, 2347013)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2157, 2347013, F2, 3, 12) (dual of [(2347013, 3), 7040882, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2157, 2796273, F2, 3, 12) (dual of [(2796273, 3), 8388662, 13]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(219, 72, F2, 3, 6) (dual of [(72, 3), 197, 7]-NRT-code), using
- extracting embedded OOA [i] based on digital (13, 19, 72)-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(219, 72, F2, 3, 6) (dual of [(72, 3), 197, 7]-NRT-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OOA(2157, 2796273, F2, 3, 12) (dual of [(2796273, 3), 8388662, 13]-NRT-code), using
(145, 145+12, large)-Net in Base 2 — Upper bound on s
There is no (145, 157, large)-net in base 2, because
- 10 times m-reduction [i] would yield (145, 147, large)-net in base 2, but