Best Known (256−21, 256, s)-Nets in Base 4
(256−21, 256, 3355450)-Net over F4 — Constructive and digital
Digital (235, 256, 3355450)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (2, 12, 10)-net over F4, using
- net from sequence [i] based on digital (2, 9)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 2 and N(F) ≥ 10, using
- net from sequence [i] based on digital (2, 9)-sequence over F4, using
- digital (223, 244, 3355440)-net over F4, using
- trace code for nets [i] based on digital (101, 122, 1677720)-net over F16, using
- net defined by OOA [i] based on linear OOA(16122, 1677720, F16, 22, 21) (dual of [(1677720, 22), 36909718, 22]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OOA(16122, 8388601, F16, 2, 21) (dual of [(8388601, 2), 16777080, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(16122, 8388602, F16, 2, 21) (dual of [(8388602, 2), 16777082, 22]-NRT-code), using
- trace code [i] based on linear OOA(25661, 4194301, F256, 2, 21) (dual of [(4194301, 2), 8388541, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25661, 8388602, F256, 21) (dual of [8388602, 8388541, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(25661, large, F256, 21) (dual of [large, large−61, 22]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [i]
- discarding factors / shortening the dual code based on linear OA(25661, large, F256, 21) (dual of [large, large−61, 22]-code), using
- OOA 2-folding [i] based on linear OA(25661, 8388602, F256, 21) (dual of [8388602, 8388541, 22]-code), using
- trace code [i] based on linear OOA(25661, 4194301, F256, 2, 21) (dual of [(4194301, 2), 8388541, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(16122, 8388602, F16, 2, 21) (dual of [(8388602, 2), 16777082, 22]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OOA(16122, 8388601, F16, 2, 21) (dual of [(8388601, 2), 16777080, 22]-NRT-code), using
- net defined by OOA [i] based on linear OOA(16122, 1677720, F16, 22, 21) (dual of [(1677720, 22), 36909718, 22]-NRT-code), using
- trace code for nets [i] based on digital (101, 122, 1677720)-net over F16, using
- digital (2, 12, 10)-net over F4, using
(256−21, 256, large)-Net over F4 — Digital
Digital (235, 256, large)-net over F4, using
- t-expansion [i] based on digital (231, 256, large)-net over F4, using
- 1 times m-reduction [i] based on digital (231, 257, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4257, large, F4, 26) (dual of [large, large−257, 27]-code), using
- 28 times code embedding in larger space [i] based on linear OA(4229, large, F4, 26) (dual of [large, large−229, 27]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,25], and designed minimum distance d ≥ |I|+1 = 27 [i]
- 28 times code embedding in larger space [i] based on linear OA(4229, large, F4, 26) (dual of [large, large−229, 27]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4257, large, F4, 26) (dual of [large, large−257, 27]-code), using
- 1 times m-reduction [i] based on digital (231, 257, large)-net over F4, using
(256−21, 256, large)-Net in Base 4 — Upper bound on s
There is no (235, 256, large)-net in base 4, because
- 19 times m-reduction [i] would yield (235, 237, large)-net in base 4, but