Best Known (168−16, 168, s)-Nets in Base 8
(168−16, 168, 2228225)-Net over F8 — Constructive and digital
Digital (152, 168, 2228225)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (38, 46, 131075)-net over F8, using
- net defined by OOA [i] based on linear OOA(846, 131075, F8, 8, 8) (dual of [(131075, 8), 1048554, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(846, 524300, F8, 8) (dual of [524300, 524254, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(846, 524302, F8, 8) (dual of [524302, 524256, 9]-code), using
- trace code [i] based on linear OA(6423, 262151, F64, 8) (dual of [262151, 262128, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(6422, 262144, F64, 8) (dual of [262144, 262122, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(6416, 262144, F64, 6) (dual of [262144, 262128, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(641, 7, F64, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- trace code [i] based on linear OA(6423, 262151, F64, 8) (dual of [262151, 262128, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(846, 524302, F8, 8) (dual of [524302, 524256, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(846, 524300, F8, 8) (dual of [524300, 524254, 9]-code), using
- net defined by OOA [i] based on linear OOA(846, 131075, F8, 8, 8) (dual of [(131075, 8), 1048554, 9]-NRT-code), using
- digital (106, 122, 2097150)-net over F8, using
- net defined by OOA [i] based on linear OOA(8122, 2097150, F8, 18, 16) (dual of [(2097150, 18), 37748578, 17]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OOA(8122, 8388601, F8, 2, 16) (dual of [(8388601, 2), 16777080, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8122, 8388602, F8, 2, 16) (dual of [(8388602, 2), 16777082, 17]-NRT-code), using
- trace code [i] based on linear OOA(6461, 4194301, F64, 2, 16) (dual of [(4194301, 2), 8388541, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6461, 8388602, F64, 16) (dual of [8388602, 8388541, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(6461, large, F64, 16) (dual of [large, large−61, 17]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- discarding factors / shortening the dual code based on linear OA(6461, large, F64, 16) (dual of [large, large−61, 17]-code), using
- OOA 2-folding [i] based on linear OA(6461, 8388602, F64, 16) (dual of [8388602, 8388541, 17]-code), using
- trace code [i] based on linear OOA(6461, 4194301, F64, 2, 16) (dual of [(4194301, 2), 8388541, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8122, 8388602, F8, 2, 16) (dual of [(8388602, 2), 16777082, 17]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OOA(8122, 8388601, F8, 2, 16) (dual of [(8388601, 2), 16777080, 17]-NRT-code), using
- net defined by OOA [i] based on linear OOA(8122, 2097150, F8, 18, 16) (dual of [(2097150, 18), 37748578, 17]-NRT-code), using
- digital (38, 46, 131075)-net over F8, using
(168−16, 168, large)-Net over F8 — Digital
Digital (152, 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
(168−16, 168, large)-Net in Base 8 — Upper bound on s
There is no (152, 168, large)-net in base 8, because
- 14 times m-reduction [i] would yield (152, 154, large)-net in base 8, but