Best Known (164−17, 164, s)-Nets in Base 4
(164−17, 164, 1048623)-Net over F4 — Constructive and digital
Digital (147, 164, 1048623)-net over F4, using
- 41 times duplication [i] based on digital (146, 163, 1048623)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (10, 18, 48)-net over F4, using
- trace code for nets [i] based on digital (1, 9, 24)-net over F16, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 1 and N(F) ≥ 24, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- trace code for nets [i] based on digital (1, 9, 24)-net over F16, 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
- digital (10, 18, 48)-net over F4, using
- (u, u+v)-construction [i] based on
(164−17, 164, 7465002)-Net over F4 — Digital
Digital (147, 164, 7465002)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4164, 7465002, F4, 17) (dual of [7465002, 7464838, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(4164, large, F4, 17) (dual of [large, large−164, 18]-code), using
- 19 times code embedding in larger space [i] 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]
- 19 times code embedding in larger space [i] based on linear OA(4145, large, F4, 17) (dual of [large, large−145, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(4164, large, F4, 17) (dual of [large, large−164, 18]-code), using
(164−17, 164, large)-Net in Base 4 — Upper bound on s
There is no (147, 164, large)-net in base 4, because
- 15 times m-reduction [i] would yield (147, 149, large)-net in base 4, but