Best Known (165, 165+15, s)-Nets in Base 2
(165, 165+15, 1198412)-Net over F2 — Constructive and digital
Digital (165, 180, 1198412)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (11, 18, 41)-net over F2, using
- digital (147, 162, 1198371)-net over F2, using
- net defined by OOA [i] based on linear OOA(2162, 1198371, F2, 15, 15) (dual of [(1198371, 15), 17975403, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2162, 8388598, F2, 15) (dual of [8388598, 8388436, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(2162, large, F2, 15) (dual of [large, large−162, 16]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(2162, large, F2, 15) (dual of [large, large−162, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2162, 8388598, F2, 15) (dual of [8388598, 8388436, 16]-code), using
- net defined by OOA [i] based on linear OOA(2162, 1198371, F2, 15, 15) (dual of [(1198371, 15), 17975403, 16]-NRT-code), using
(165, 165+15, 1677761)-Net over F2 — Digital
Digital (165, 180, 1677761)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2180, 1677761, F2, 5, 15) (dual of [(1677761, 5), 8388625, 16]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(218, 41, F2, 5, 7) (dual of [(41, 5), 187, 8]-NRT-code), using
- extracting embedded OOA [i] based on digital (11, 18, 41)-net over F2, using
- linear OOA(2162, 1677720, F2, 5, 15) (dual of [(1677720, 5), 8388438, 16]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2162, 8388600, F2, 15) (dual of [8388600, 8388438, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(2162, large, F2, 15) (dual of [large, large−162, 16]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(2162, large, F2, 15) (dual of [large, large−162, 16]-code), using
- OOA 5-folding [i] based on linear OA(2162, 8388600, F2, 15) (dual of [8388600, 8388438, 16]-code), using
- linear OOA(218, 41, F2, 5, 7) (dual of [(41, 5), 187, 8]-NRT-code), using
- (u, u+v)-construction [i] based on
(165, 165+15, large)-Net in Base 2 — Upper bound on s
There is no (165, 180, large)-net in base 2, because
- 13 times m-reduction [i] would yield (165, 167, large)-net in base 2, but