Best Known (164, 164+14, s)-Nets in Base 2
(164, 164+14, 1198405)-Net over F2 — Constructive and digital
Digital (164, 178, 1198405)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (10, 17, 34)-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
(164, 164+14, 2097184)-Net over F2 — Digital
Digital (164, 178, 2097184)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2178, 2097184, F2, 4, 14) (dual of [(2097184, 4), 8388558, 15]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(217, 34, F2, 4, 7) (dual of [(34, 4), 119, 8]-NRT-code), using
- extracting embedded OOA [i] based on digital (10, 17, 34)-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(217, 34, F2, 4, 7) (dual of [(34, 4), 119, 8]-NRT-code), using
- (u, u+v)-construction [i] based on
(164, 164+14, large)-Net in Base 2 — Upper bound on s
There is no (164, 178, large)-net in base 2, because
- 12 times m-reduction [i] would yield (164, 166, large)-net in base 2, but