Best Known (212, 229, s)-Nets in Base 4
(212, 229, 4195328)-Net over F4 — Constructive and digital
Digital (212, 229, 4195328)-net over F4, using
- 41 times duplication [i] based on digital (211, 228, 4195328)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (24, 32, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 8, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 8, 257)-net over F256, 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 (24, 32, 1028)-net over F4, using
- (u, u+v)-construction [i] based on
(212, 229, large)-Net over F4 — Digital
Digital (212, 229, large)-net over F4, using
- 42 times duplication [i] based on digital (210, 227, large)-net over F4, using
- t-expansion [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
- t-expansion [i] based on digital (204, 227, large)-net over F4, using
(212, 229, large)-Net in Base 4 — Upper bound on s
There is no (212, 229, large)-net in base 4, because
- 15 times m-reduction [i] would yield (212, 214, large)-net in base 4, but