Best Known (173−18, 173, s)-Nets in Base 8
(173−18, 173, 1866183)-Net over F8 — Constructive and digital
Digital (155, 173, 1866183)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (26, 35, 2049)-net over F8, using
- net defined by OOA [i] based on linear OOA(835, 2049, F8, 9, 9) (dual of [(2049, 9), 18406, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(835, 8197, F8, 9) (dual of [8197, 8162, 10]-code), using
- 1 times code embedding in larger space [i] based on linear OA(834, 8196, F8, 9) (dual of [8196, 8162, 10]-code), using
- trace code [i] based on linear OA(6417, 4098, F64, 9) (dual of [4098, 4081, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(6417, 4096, F64, 9) (dual of [4096, 4079, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(6415, 4096, F64, 8) (dual of [4096, 4081, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(640, 2, F64, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- trace code [i] based on linear OA(6417, 4098, F64, 9) (dual of [4098, 4081, 10]-code), using
- 1 times code embedding in larger space [i] based on linear OA(834, 8196, F8, 9) (dual of [8196, 8162, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(835, 8197, F8, 9) (dual of [8197, 8162, 10]-code), using
- net defined by OOA [i] based on linear OOA(835, 2049, F8, 9, 9) (dual of [(2049, 9), 18406, 10]-NRT-code), using
- digital (120, 138, 1864134)-net over F8, using
- trace code for nets [i] based on digital (51, 69, 932067)-net over F64, using
- net defined by OOA [i] based on linear OOA(6469, 932067, F64, 18, 18) (dual of [(932067, 18), 16777137, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(6469, large, F64, 18) (dual of [large, large−69, 19]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- OA 9-folding and stacking [i] based on linear OA(6469, large, F64, 18) (dual of [large, large−69, 19]-code), using
- net defined by OOA [i] based on linear OOA(6469, 932067, F64, 18, 18) (dual of [(932067, 18), 16777137, 19]-NRT-code), using
- trace code for nets [i] based on digital (51, 69, 932067)-net over F64, using
- digital (26, 35, 2049)-net over F8, using
(173−18, 173, large)-Net over F8 — Digital
Digital (155, 173, large)-net over F8, using
- 85 times duplication [i] based on digital (150, 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
(173−18, 173, large)-Net in Base 8 — Upper bound on s
There is no (155, 173, large)-net in base 8, because
- 16 times m-reduction [i] would yield (155, 157, large)-net in base 8, but