Best Known (157−17, 157, s)-Nets in Base 4
(157−17, 157, 1048592)-Net over F4 — Constructive and digital
Digital (140, 157, 1048592)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (4, 12, 17)-net over F4, using
- digital (128, 145, 1048575)-net over F4, using
- net defined by OOA [i] based on linear OOA(4145, 1048575, F4, 17, 17) (dual of [(1048575, 17), 17825630, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(4145, 8388601, F4, 17) (dual of [8388601, 8388456, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(4145, large, F4, 17) (dual of [large, large−145, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 424−1, 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(4145, large, F4, 17) (dual of [large, large−145, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(4145, 8388601, F4, 17) (dual of [8388601, 8388456, 18]-code), using
- net defined by OOA [i] based on linear OOA(4145, 1048575, F4, 17, 17) (dual of [(1048575, 17), 17825630, 18]-NRT-code), using
(157−17, 157, 4194318)-Net over F4 — Digital
Digital (140, 157, 4194318)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4157, 4194318, F4, 2, 17) (dual of [(4194318, 2), 8388479, 18]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(412, 17, F4, 2, 8) (dual of [(17, 2), 22, 9]-NRT-code), using
- extracting embedded OOA [i] based on digital (4, 12, 17)-net over F4, using
- linear OOA(4145, 4194301, F4, 2, 17) (dual of [(4194301, 2), 8388457, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4145, 8388602, F4, 17) (dual of [8388602, 8388457, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(4145, large, F4, 17) (dual of [large, large−145, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 424−1, 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(4145, large, F4, 17) (dual of [large, large−145, 18]-code), using
- OOA 2-folding [i] based on linear OA(4145, 8388602, F4, 17) (dual of [8388602, 8388457, 18]-code), using
- linear OOA(412, 17, F4, 2, 8) (dual of [(17, 2), 22, 9]-NRT-code), using
- (u, u+v)-construction [i] based on
(157−17, 157, large)-Net in Base 4 — Upper bound on s
There is no (140, 157, large)-net in base 4, because
- 15 times m-reduction [i] would yield (140, 142, large)-net in base 4, but