Best Known (168−23, 168, s)-Nets in Base 8
(168−23, 168, 762600)-Net over F8 — Constructive and digital
Digital (145, 168, 762600)-net over F8, using
- 87 times duplication [i] based on digital (138, 161, 762600)-net over F8, using
- net defined by OOA [i] based on linear OOA(8161, 762600, F8, 23, 23) (dual of [(762600, 23), 17539639, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8161, 8388601, F8, 23) (dual of [8388601, 8388440, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(8161, large, F8, 23) (dual of [large, large−161, 24]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 816−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(8161, large, F8, 23) (dual of [large, large−161, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8161, 8388601, F8, 23) (dual of [8388601, 8388440, 24]-code), using
- net defined by OOA [i] based on linear OOA(8161, 762600, F8, 23, 23) (dual of [(762600, 23), 17539639, 24]-NRT-code), using
(168−23, 168, large)-Net over F8 — Digital
Digital (145, 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
(168−23, 168, large)-Net in Base 8 — Upper bound on s
There is no (145, 168, large)-net in base 8, because
- 21 times m-reduction [i] would yield (145, 147, large)-net in base 8, but