Best Known (140−15, 140, s)-Nets in Base 8
(140−15, 140, 2399473)-Net over F8 — Constructive and digital
Digital (125, 140, 2399473)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (19, 26, 2731)-net over F8, using
- net defined by OOA [i] based on linear OOA(826, 2731, F8, 7, 7) (dual of [(2731, 7), 19091, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(826, 8194, F8, 7) (dual of [8194, 8168, 8]-code), using
- trace code [i] based on linear OA(6413, 4097, F64, 7) (dual of [4097, 4084, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- trace code [i] based on linear OA(6413, 4097, F64, 7) (dual of [4097, 4084, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(826, 8194, F8, 7) (dual of [8194, 8168, 8]-code), using
- net defined by OOA [i] based on linear OOA(826, 2731, F8, 7, 7) (dual of [(2731, 7), 19091, 8]-NRT-code), using
- digital (99, 114, 2396742)-net over F8, using
- trace code for nets [i] based on digital (42, 57, 1198371)-net over F64, using
- net defined by OOA [i] based on linear OOA(6457, 1198371, F64, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(6457, 8388598, F64, 15) (dual of [8388598, 8388541, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(6457, large, F64, 15) (dual of [large, large−57, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(6457, large, F64, 15) (dual of [large, large−57, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(6457, 8388598, F64, 15) (dual of [8388598, 8388541, 16]-code), using
- net defined by OOA [i] based on linear OOA(6457, 1198371, F64, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
- trace code for nets [i] based on digital (42, 57, 1198371)-net over F64, using
- digital (19, 26, 2731)-net over F8, using
(140−15, 140, large)-Net over F8 — Digital
Digital (125, 140, large)-net over F8, using
- 6 times m-reduction [i] based on digital (125, 146, large)-net over F8, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8146, large, F8, 21) (dual of [large, large−146, 22]-code), using
- 1 times code embedding in larger space [i] based on linear OA(8145, large, F8, 21) (dual of [large, large−145, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 816−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- 1 times code embedding in larger space [i] based on linear OA(8145, large, F8, 21) (dual of [large, large−145, 22]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8146, large, F8, 21) (dual of [large, large−146, 22]-code), using
(140−15, 140, large)-Net in Base 8 — Upper bound on s
There is no (125, 140, large)-net in base 8, because
- 13 times m-reduction [i] would yield (125, 127, large)-net in base 8, but