Best Known (210, 210+16, s)-Nets in Base 4
(210, 210+16, 4198396)-Net over F4 — Constructive and digital
Digital (210, 226, 4198396)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (34, 42, 4096)-net over F4, using
- net defined by OOA [i] based on linear OOA(442, 4096, F4, 8, 8) (dual of [(4096, 8), 32726, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(442, 16384, F4, 8) (dual of [16384, 16342, 9]-code), using
- 1 times truncation [i] based on linear OA(443, 16385, F4, 9) (dual of [16385, 16342, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, 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(443, 16385, F4, 9) (dual of [16385, 16342, 10]-code), using
- OA 4-folding and stacking [i] based on linear OA(442, 16384, F4, 8) (dual of [16384, 16342, 9]-code), using
- net defined by OOA [i] based on linear OOA(442, 4096, F4, 8, 8) (dual of [(4096, 8), 32726, 9]-NRT-code), using
- digital (168, 184, 4194300)-net over F4, using
- trace code for nets [i] based on digital (76, 92, 2097150)-net over F16, using
- net defined by OOA [i] based on linear OOA(1692, 2097150, F16, 18, 16) (dual of [(2097150, 18), 37748608, 17]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OOA(1692, 8388601, F16, 2, 16) (dual of [(8388601, 2), 16777110, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(1692, 8388602, F16, 2, 16) (dual of [(8388602, 2), 16777112, 17]-NRT-code), using
- trace code [i] based on linear OOA(25646, 4194301, F256, 2, 16) (dual of [(4194301, 2), 8388556, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25646, 8388602, F256, 16) (dual of [8388602, 8388556, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(25646, large, F256, 16) (dual of [large, large−46, 17]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- discarding factors / shortening the dual code based on linear OA(25646, large, F256, 16) (dual of [large, large−46, 17]-code), using
- OOA 2-folding [i] based on linear OA(25646, 8388602, F256, 16) (dual of [8388602, 8388556, 17]-code), using
- trace code [i] based on linear OOA(25646, 4194301, F256, 2, 16) (dual of [(4194301, 2), 8388556, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(1692, 8388602, F16, 2, 16) (dual of [(8388602, 2), 16777112, 17]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OOA(1692, 8388601, F16, 2, 16) (dual of [(8388601, 2), 16777110, 17]-NRT-code), using
- net defined by OOA [i] based on linear OOA(1692, 2097150, F16, 18, 16) (dual of [(2097150, 18), 37748608, 17]-NRT-code), using
- trace code for nets [i] based on digital (76, 92, 2097150)-net over F16, using
- digital (34, 42, 4096)-net over F4, using
(210, 210+16, large)-Net over F4 — Digital
Digital (210, 226, large)-net over F4, using
- t-expansion [i] based on digital (204, 226, large)-net over F4, using
- 1 times m-reduction [i] based on digital (204, 227, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4227, large, F4, 23) (dual of [large, large−227, 24]-code), using
- 22 times code embedding in larger space [i] based on linear OA(4205, large, F4, 23) (dual of [large, large−205, 24]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,22], and designed minimum distance d ≥ |I|+1 = 24 [i]
- 22 times code embedding in larger space [i] based on linear OA(4205, large, F4, 23) (dual of [large, large−205, 24]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4227, large, F4, 23) (dual of [large, large−227, 24]-code), using
- 1 times m-reduction [i] based on digital (204, 227, large)-net over F4, using
(210, 210+16, large)-Net in Base 4 — Upper bound on s
There is no (210, 226, large)-net in base 4, because
- 14 times m-reduction [i] would yield (210, 212, large)-net in base 4, but