Best Known (87−10, 87, s)-Nets in Base 8
(87−10, 87, 3357456)-Net over F8 — Constructive and digital
Digital (77, 87, 3357456)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (8, 13, 2016)-net over F8, using
- net defined by OOA [i] based on linear OOA(813, 2016, F8, 5, 5) (dual of [(2016, 5), 10067, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(813, 4033, F8, 5) (dual of [4033, 4020, 6]-code), using
- net defined by OOA [i] based on linear OOA(813, 2016, F8, 5, 5) (dual of [(2016, 5), 10067, 6]-NRT-code), using
- digital (64, 74, 3355440)-net over F8, using
- trace code for nets [i] based on digital (27, 37, 1677720)-net over F64, using
- net defined by OOA [i] based on linear OOA(6437, 1677720, F64, 10, 10) (dual of [(1677720, 10), 16777163, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(6437, 8388600, F64, 10) (dual of [8388600, 8388563, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(6437, large, F64, 10) (dual of [large, large−37, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(6437, large, F64, 10) (dual of [large, large−37, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(6437, 8388600, F64, 10) (dual of [8388600, 8388563, 11]-code), using
- net defined by OOA [i] based on linear OOA(6437, 1677720, F64, 10, 10) (dual of [(1677720, 10), 16777163, 11]-NRT-code), using
- trace code for nets [i] based on digital (27, 37, 1677720)-net over F64, using
- digital (8, 13, 2016)-net over F8, using
(87−10, 87, large)-Net over F8 — Digital
Digital (77, 87, large)-net over F8, using
- t-expansion [i] based on digital (76, 87, large)-net over F8, using
- 2 times m-reduction [i] based on digital (76, 89, large)-net over F8, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(889, large, F8, 13) (dual of [large, large−89, 14]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 88−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(889, large, F8, 13) (dual of [large, large−89, 14]-code), using
- 2 times m-reduction [i] based on digital (76, 89, large)-net over F8, using
(87−10, 87, large)-Net in Base 8 — Upper bound on s
There is no (77, 87, large)-net in base 8, because
- 8 times m-reduction [i] would yield (77, 79, large)-net in base 8, but