Best Known (165, 165+14, s)-Nets in Base 2
(165, 165+14, 1198412)-Net over F2 — Constructive and digital
Digital (165, 179, 1198412)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (11, 18, 41)-net over F2, using
- digital (147, 161, 1198371)-net over F2, using
- net defined by OOA [i] based on linear OOA(2161, 1198371, F2, 14, 14) (dual of [(1198371, 14), 16777033, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(2161, 8388597, F2, 14) (dual of [8388597, 8388436, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(2161, large, F2, 14) (dual of [large, large−161, 15]-code), using
- the primitive narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(2161, large, F2, 14) (dual of [large, large−161, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(2161, 8388597, F2, 14) (dual of [8388597, 8388436, 15]-code), using
- net defined by OOA [i] based on linear OOA(2161, 1198371, F2, 14, 14) (dual of [(1198371, 14), 16777033, 15]-NRT-code), using
(165, 165+14, 2097191)-Net over F2 — Digital
Digital (165, 179, 2097191)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2179, 2097191, F2, 4, 14) (dual of [(2097191, 4), 8388585, 15]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(218, 41, F2, 4, 7) (dual of [(41, 4), 146, 8]-NRT-code), using
- extracting embedded OOA [i] based on digital (11, 18, 41)-net over F2, using
- linear OOA(2161, 2097150, F2, 4, 14) (dual of [(2097150, 4), 8388439, 15]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2161, 8388600, F2, 14) (dual of [8388600, 8388439, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(2161, large, F2, 14) (dual of [large, large−161, 15]-code), using
- the primitive narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(2161, large, F2, 14) (dual of [large, large−161, 15]-code), using
- OOA 4-folding [i] based on linear OA(2161, 8388600, F2, 14) (dual of [8388600, 8388439, 15]-code), using
- linear OOA(218, 41, F2, 4, 7) (dual of [(41, 4), 146, 8]-NRT-code), using
- (u, u+v)-construction [i] based on
(165, 165+14, large)-Net in Base 2 — Upper bound on s
There is no (165, 179, large)-net in base 2, because
- 12 times m-reduction [i] would yield (165, 167, large)-net in base 2, but