Best Known (61, 69, s)-Nets in Base 8
(61, 69, 4196316)-Net over F8 — Constructive and digital
Digital (61, 69, 4196316)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (7, 11, 2016)-net over F8, using
- net defined by OOA [i] based on linear OOA(811, 2016, F8, 4, 4) (dual of [(2016, 4), 8053, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(811, 2016, F8, 3, 4) (dual of [(2016, 3), 6037, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(811, 4032, F8, 4) (dual of [4032, 4021, 5]-code), using
- base reduction for projective spaces (embedding PG(5,64) in PG(10,8)) [i] based on linear OA(646, 4032, F64, 4) (dual of [4032, 4026, 5]-code), using
- 1 times truncation [i] based on linear OA(647, 4033, F64, 5) (dual of [4033, 4026, 6]-code), using
- base reduction for projective spaces (embedding PG(5,64) in PG(10,8)) [i] based on linear OA(646, 4032, F64, 4) (dual of [4032, 4026, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(811, 4032, F8, 4) (dual of [4032, 4021, 5]-code), using
- appending kth column [i] based on linear OOA(811, 2016, F8, 3, 4) (dual of [(2016, 3), 6037, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(811, 2016, F8, 4, 4) (dual of [(2016, 4), 8053, 5]-NRT-code), using
- digital (50, 58, 4194300)-net over F8, using
- net defined by OOA [i] based on linear OOA(858, 4194300, F8, 10, 8) (dual of [(4194300, 10), 41942942, 9]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(858, 8388601, F8, 2, 8) (dual of [(8388601, 2), 16777144, 9]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(858, 8388602, F8, 2, 8) (dual of [(8388602, 2), 16777146, 9]-NRT-code), using
- trace code [i] based on linear OOA(6429, 4194301, F64, 2, 8) (dual of [(4194301, 2), 8388573, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6429, 8388602, F64, 8) (dual of [8388602, 8388573, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(6429, large, F64, 8) (dual of [large, large−29, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(6429, large, F64, 8) (dual of [large, large−29, 9]-code), using
- OOA 2-folding [i] based on linear OA(6429, 8388602, F64, 8) (dual of [8388602, 8388573, 9]-code), using
- trace code [i] based on linear OOA(6429, 4194301, F64, 2, 8) (dual of [(4194301, 2), 8388573, 9]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(858, 8388602, F8, 2, 8) (dual of [(8388602, 2), 16777146, 9]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(858, 8388601, F8, 2, 8) (dual of [(8388601, 2), 16777144, 9]-NRT-code), using
- net defined by OOA [i] based on linear OOA(858, 4194300, F8, 10, 8) (dual of [(4194300, 10), 41942942, 9]-NRT-code), using
- digital (7, 11, 2016)-net over F8, using
(61, 69, large)-Net over F8 — Digital
Digital (61, 69, large)-net over F8, using
- 84 times duplication [i] based on digital (57, 65, large)-net over F8, using
- t-expansion [i] based on digital (55, 65, large)-net over F8, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(865, large, F8, 10) (dual of [large, large−65, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 88−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(865, large, F8, 10) (dual of [large, large−65, 11]-code), using
- t-expansion [i] based on digital (55, 65, large)-net over F8, using
(61, 69, large)-Net in Base 8 — Upper bound on s
There is no (61, 69, large)-net in base 8, because
- 6 times m-reduction [i] would yield (61, 63, large)-net in base 8, but