Best Known (138, 138+12, s)-Nets in Base 2
(138, 138+12, 1398121)-Net over F2 — Constructive and digital
Digital (138, 150, 1398121)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (6, 12, 21)-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
(138, 138+12, 2097171)-Net over F2 — Digital
Digital (138, 150, 2097171)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2150, 2097171, F2, 4, 12) (dual of [(2097171, 4), 8388534, 13]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(212, 21, F2, 4, 6) (dual of [(21, 4), 72, 7]-NRT-code), using
- extracting embedded OOA [i] based on digital (6, 12, 21)-net over F2, using
- linear OOA(2138, 2097150, F2, 4, 12) (dual of [(2097150, 4), 8388462, 13]-NRT-code), using
- OOA 4-folding [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
- OOA 4-folding [i] based on linear OA(2138, 8388600, F2, 12) (dual of [8388600, 8388462, 13]-code), using
- linear OOA(212, 21, F2, 4, 6) (dual of [(21, 4), 72, 7]-NRT-code), using
- (u, u+v)-construction [i] based on
(138, 138+12, large)-Net in Base 2 — Upper bound on s
There is no (138, 150, large)-net in base 2, because
- 10 times m-reduction [i] would yield (138, 140, large)-net in base 2, but