Best Known (227, 227+17, s)-Nets in Base 4
(227, 227+17, 4210684)-Net over F4 — Constructive and digital
Digital (227, 244, 4210684)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (40, 48, 16384)-net over F4, using
- net defined by OOA [i] based on linear OOA(448, 16384, F4, 8, 8) (dual of [(16384, 8), 131024, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(448, 65536, F4, 8) (dual of [65536, 65488, 9]-code), using
- 1 times truncation [i] based on linear OA(449, 65537, F4, 9) (dual of [65537, 65488, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, 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(449, 65537, F4, 9) (dual of [65537, 65488, 10]-code), using
- OA 4-folding and stacking [i] based on linear OA(448, 65536, F4, 8) (dual of [65536, 65488, 9]-code), using
- net defined by OOA [i] based on linear OOA(448, 16384, F4, 8, 8) (dual of [(16384, 8), 131024, 9]-NRT-code), using
- digital (179, 196, 4194300)-net over F4, using
- trace code for nets [i] based on digital (81, 98, 2097150)-net over F16, using
- net defined by OOA [i] based on linear OOA(1698, 2097150, F16, 18, 17) (dual of [(2097150, 18), 37748602, 18]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OOA(1698, 8388601, F16, 2, 17) (dual of [(8388601, 2), 16777104, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(1698, 8388602, F16, 2, 17) (dual of [(8388602, 2), 16777106, 18]-NRT-code), using
- trace code [i] based on linear OOA(25649, 4194301, F256, 2, 17) (dual of [(4194301, 2), 8388553, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25649, 8388602, F256, 17) (dual of [8388602, 8388553, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(25649, large, F256, 17) (dual of [large, large−49, 18]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(25649, large, F256, 17) (dual of [large, large−49, 18]-code), using
- OOA 2-folding [i] based on linear OA(25649, 8388602, F256, 17) (dual of [8388602, 8388553, 18]-code), using
- trace code [i] based on linear OOA(25649, 4194301, F256, 2, 17) (dual of [(4194301, 2), 8388553, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(1698, 8388602, F16, 2, 17) (dual of [(8388602, 2), 16777106, 18]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OOA(1698, 8388601, F16, 2, 17) (dual of [(8388601, 2), 16777104, 18]-NRT-code), using
- net defined by OOA [i] based on linear OOA(1698, 2097150, F16, 18, 17) (dual of [(2097150, 18), 37748602, 18]-NRT-code), using
- trace code for nets [i] based on digital (81, 98, 2097150)-net over F16, using
- digital (40, 48, 16384)-net over F4, using
(227, 227+17, large)-Net over F4 — Digital
Digital (227, 244, large)-net over F4, using
- t-expansion [i] based on digital (222, 244, large)-net over F4, using
- 3 times m-reduction [i] based on digital (222, 247, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4247, large, F4, 25) (dual of [large, large−247, 26]-code), using
- 30 times code embedding in larger space [i] based on linear OA(4217, large, F4, 25) (dual of [large, large−217, 26]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 424−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- 30 times code embedding in larger space [i] based on linear OA(4217, large, F4, 25) (dual of [large, large−217, 26]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4247, large, F4, 25) (dual of [large, large−247, 26]-code), using
- 3 times m-reduction [i] based on digital (222, 247, large)-net over F4, using
(227, 227+17, large)-Net in Base 4 — Upper bound on s
There is no (227, 244, large)-net in base 4, because
- 15 times m-reduction [i] would yield (227, 229, large)-net in base 4, but