Best Known (228, 228+17, s)-Nets in Base 2
(228, 228+17, 1056767)-Net over F2 — Constructive and digital
Digital (228, 245, 1056767)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (52, 60, 8192)-net over F2, using
- net defined by OOA [i] based on linear OOA(260, 8192, F2, 8, 8) (dual of [(8192, 8), 65476, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(260, 32768, F2, 8) (dual of [32768, 32708, 9]-code), using
- 1 times truncation [i] based on linear OA(261, 32769, F2, 9) (dual of [32769, 32708, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 32769 | 230−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(261, 32769, F2, 9) (dual of [32769, 32708, 10]-code), using
- OA 4-folding and stacking [i] based on linear OA(260, 32768, F2, 8) (dual of [32768, 32708, 9]-code), using
- net defined by OOA [i] based on linear OOA(260, 8192, F2, 8, 8) (dual of [(8192, 8), 65476, 9]-NRT-code), using
- digital (168, 185, 1048575)-net over F2, using
- net defined by OOA [i] based on linear OOA(2185, 1048575, F2, 17, 17) (dual of [(1048575, 17), 17825590, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2185, 8388601, F2, 17) (dual of [8388601, 8388416, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2185, large, F2, 17) (dual of [large, large−185, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 8388609 | 246−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2185, large, F2, 17) (dual of [large, large−185, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2185, 8388601, F2, 17) (dual of [8388601, 8388416, 18]-code), using
- net defined by OOA [i] based on linear OOA(2185, 1048575, F2, 17, 17) (dual of [(1048575, 17), 17825590, 18]-NRT-code), using
- digital (52, 60, 8192)-net over F2, using
(228, 228+17, 2276102)-Net over F2 — Digital
Digital (228, 245, 2276102)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2245, 2276102, F2, 3, 17) (dual of [(2276102, 3), 6828061, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2245, 2807123, F2, 3, 17) (dual of [(2807123, 3), 8421124, 18]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(260, 10922, F2, 3, 8) (dual of [(10922, 3), 32706, 9]-NRT-code), using
- OOA 3-folding [i] based on linear OA(260, 32766, F2, 8) (dual of [32766, 32706, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(260, 32768, F2, 8) (dual of [32768, 32708, 9]-code), using
- 1 times truncation [i] based on linear OA(261, 32769, F2, 9) (dual of [32769, 32708, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 32769 | 230−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(261, 32769, F2, 9) (dual of [32769, 32708, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(260, 32768, F2, 8) (dual of [32768, 32708, 9]-code), using
- OOA 3-folding [i] based on linear OA(260, 32766, F2, 8) (dual of [32766, 32706, 9]-code), using
- linear OOA(2185, 2796201, F2, 3, 17) (dual of [(2796201, 3), 8388418, 18]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2185, large, F2, 17) (dual of [large, large−185, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 8388609 | 246−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- OOA 3-folding [i] based on linear OA(2185, large, F2, 17) (dual of [large, large−185, 18]-code), using
- linear OOA(260, 10922, F2, 3, 8) (dual of [(10922, 3), 32706, 9]-NRT-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OOA(2245, 2807123, F2, 3, 17) (dual of [(2807123, 3), 8421124, 18]-NRT-code), using
(228, 228+17, large)-Net in Base 2 — Upper bound on s
There is no (228, 245, large)-net in base 2, because
- 15 times m-reduction [i] would yield (228, 230, large)-net in base 2, but