Best Known (51, 63, s)-Nets in Base 256
(51, 63, 4194557)-Net over F256 — Constructive and digital
Digital (51, 63, 4194557)-net over F256, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 3, 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
- digital (6, 10, 1398100)-net over F256, using
- s-reduction based on digital (6, 10, 4194301)-net over F256, using
- net defined by OOA [i] based on linear OOA(25610, 4194301, F256, 4, 4) (dual of [(4194301, 4), 16777194, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(25610, 8388602, F256, 4) (dual of [8388602, 8388592, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(25610, large, F256, 4) (dual of [large, large−10, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(25610, large, F256, 4) (dual of [large, large−10, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(25610, 8388602, F256, 4) (dual of [8388602, 8388592, 5]-code), using
- net defined by OOA [i] based on linear OOA(25610, 4194301, F256, 4, 4) (dual of [(4194301, 4), 16777194, 5]-NRT-code), using
- s-reduction based on digital (6, 10, 4194301)-net over F256, using
- digital (10, 16, 1398100)-net over F256, using
- s-reduction based on digital (10, 16, 2796201)-net over F256, using
- net defined by OOA [i] based on linear OOA(25616, 2796201, F256, 6, 6) (dual of [(2796201, 6), 16777190, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(25616, large, F256, 6) (dual of [large, large−16, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- OA 3-folding and stacking [i] based on linear OA(25616, large, F256, 6) (dual of [large, large−16, 7]-code), using
- net defined by OOA [i] based on linear OOA(25616, 2796201, F256, 6, 6) (dual of [(2796201, 6), 16777190, 7]-NRT-code), using
- s-reduction based on digital (10, 16, 2796201)-net over F256, using
- digital (22, 34, 1398100)-net over F256, using
- net defined by OOA [i] based on linear OOA(25634, 1398100, F256, 12, 12) (dual of [(1398100, 12), 16777166, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(25634, 8388600, F256, 12) (dual of [8388600, 8388566, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(25634, large, F256, 12) (dual of [large, large−34, 13]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(25634, large, F256, 12) (dual of [large, large−34, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(25634, 8388600, F256, 12) (dual of [8388600, 8388566, 13]-code), using
- net defined by OOA [i] based on linear OOA(25634, 1398100, F256, 12, 12) (dual of [(1398100, 12), 16777166, 13]-NRT-code), using
- digital (0, 3, 257)-net over F256, using
(51, 63, large)-Net over F256 — Digital
Digital (51, 63, large)-net over F256, using
- t-expansion [i] based on digital (47, 63, large)-net over F256, using
- 5 times m-reduction [i] based on digital (47, 68, large)-net over F256, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(25668, large, F256, 21) (dual of [large, large−68, 22]-code), using
- 7 times code embedding in larger space [i] 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]
- 7 times code embedding in larger space [i] based on linear OA(25661, large, F256, 21) (dual of [large, large−61, 22]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(25668, large, F256, 21) (dual of [large, large−68, 22]-code), using
- 5 times m-reduction [i] based on digital (47, 68, large)-net over F256, using
(51, 63, large)-Net in Base 256 — Upper bound on s
There is no (51, 63, large)-net in base 256, because
- 10 times m-reduction [i] would yield (51, 53, large)-net in base 256, but