Best Known (157, 157+14, s)-Nets in Base 8
(157, 157+14, 4809870)-Net over F8 — Constructive and digital
Digital (157, 171, 4809870)-net over F8, using
- generalized (u, u+v)-construction [i] based on
- digital (12, 16, 16386)-net over F8, using
- net defined by OOA [i] based on linear OOA(816, 16386, F8, 4, 4) (dual of [(16386, 4), 65528, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(816, 16386, F8, 3, 4) (dual of [(16386, 3), 49142, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(816, 32772, F8, 4) (dual of [32772, 32756, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(816, 32773, F8, 4) (dual of [32773, 32757, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(816, 32768, F8, 4) (dual of [32768, 32752, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(811, 32768, F8, 3) (dual of [32768, 32757, 4]-code or 32768-cap in PG(10,8)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(80, 5, F8, 0) (dual of [5, 5, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- discarding factors / shortening the dual code based on linear OA(816, 32773, F8, 4) (dual of [32773, 32757, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(816, 32772, F8, 4) (dual of [32772, 32756, 5]-code), using
- appending kth column [i] based on linear OOA(816, 16386, F8, 3, 4) (dual of [(16386, 3), 49142, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(816, 16386, F8, 4, 4) (dual of [(16386, 4), 65528, 5]-NRT-code), using
- digital (42, 49, 2396742)-net over F8, using
- s-reduction based on digital (42, 49, 2796200)-net over F8, using
- net defined by OOA [i] based on linear OOA(849, 2796200, F8, 7, 7) (dual of [(2796200, 7), 19573351, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(849, 8388601, F8, 7) (dual of [8388601, 8388552, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(849, large, F8, 7) (dual of [large, large−49, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 816−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(849, large, F8, 7) (dual of [large, large−49, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(849, 8388601, F8, 7) (dual of [8388601, 8388552, 8]-code), using
- net defined by OOA [i] based on linear OOA(849, 2796200, F8, 7, 7) (dual of [(2796200, 7), 19573351, 8]-NRT-code), using
- s-reduction based on digital (42, 49, 2796200)-net over F8, using
- digital (92, 106, 2396742)-net over F8, using
- trace code for nets [i] based on digital (39, 53, 1198371)-net over F64, using
- net defined by OOA [i] based on linear OOA(6453, 1198371, F64, 14, 14) (dual of [(1198371, 14), 16777141, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(6453, 8388597, F64, 14) (dual of [8388597, 8388544, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(6453, large, F64, 14) (dual of [large, large−53, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(6453, large, F64, 14) (dual of [large, large−53, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(6453, 8388597, F64, 14) (dual of [8388597, 8388544, 15]-code), using
- net defined by OOA [i] based on linear OOA(6453, 1198371, F64, 14, 14) (dual of [(1198371, 14), 16777141, 15]-NRT-code), using
- trace code for nets [i] based on digital (39, 53, 1198371)-net over F64, using
- digital (12, 16, 16386)-net over F8, using
(157, 157+14, large)-Net over F8 — Digital
Digital (157, 171, large)-net over F8, using
- 83 times duplication [i] based on digital (154, 168, large)-net over F8, using
- t-expansion [i] based on digital (144, 168, large)-net over F8, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8168, large, F8, 24) (dual of [large, large−168, 25]-code), using
- the primitive narrow-sense BCH-code C(I) with length 16777215 = 88−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8168, large, F8, 24) (dual of [large, large−168, 25]-code), using
- t-expansion [i] based on digital (144, 168, large)-net over F8, using
(157, 157+14, large)-Net in Base 8 — Upper bound on s
There is no (157, 171, large)-net in base 8, because
- 12 times m-reduction [i] would yield (157, 159, large)-net in base 8, but