Best Known (125−9, 125, s)-Nets in Base 8
(125−9, 125, large)-Net over F8 — Constructive and digital
Digital (116, 125, large)-net over F8, using
- 2 times m-reduction [i] based on digital (116, 127, large)-net over F8, using
- generalized (u, u+v)-construction [i] based on
- digital (9, 12, 2931457)-net over F8, using
- net defined by OOA [i] based on linear OOA(812, 2931457, F8, 3, 3) (dual of [(2931457, 3), 8794359, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(812, 2931457, F8, 2, 3) (dual of [(2931457, 2), 5862902, 4]-NRT-code), using
- OAs with strength 3, b ≠ 2, and m > 3 are always embeddable [i] based on linear OA(812, 2931457, F8, 3) (dual of [2931457, 2931445, 4]-code or 2931457-cap in PG(11,8)), using
- appending kth column [i] based on linear OOA(812, 2931457, F8, 2, 3) (dual of [(2931457, 2), 5862902, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(812, 2931457, F8, 3, 3) (dual of [(2931457, 3), 8794359, 4]-NRT-code), using
- digital (28, 33, 3355440)-net over F8, using
- s-reduction based on digital (28, 33, 4194301)-net over F8, using
- net defined by OOA [i] based on linear OOA(833, 4194301, F8, 5, 5) (dual of [(4194301, 5), 20971472, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(833, large, F8, 5) (dual of [large, large−33, 6]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 816−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- OOA 2-folding and stacking with additional row [i] based on linear OA(833, large, F8, 5) (dual of [large, large−33, 6]-code), using
- net defined by OOA [i] based on linear OOA(833, 4194301, F8, 5, 5) (dual of [(4194301, 5), 20971472, 6]-NRT-code), using
- s-reduction based on digital (28, 33, 4194301)-net over F8, using
- digital (71, 82, 3355440)-net over F8, using
- trace code for nets [i] based on digital (30, 41, 1677720)-net over F64, using
- net defined by OOA [i] based on linear OOA(6441, 1677720, F64, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(6441, 8388601, F64, 11) (dual of [8388601, 8388560, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(6441, large, F64, 11) (dual of [large, large−41, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(6441, large, F64, 11) (dual of [large, large−41, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(6441, 8388601, F64, 11) (dual of [8388601, 8388560, 12]-code), using
- net defined by OOA [i] based on linear OOA(6441, 1677720, F64, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
- trace code for nets [i] based on digital (30, 41, 1677720)-net over F64, using
- digital (9, 12, 2931457)-net over F8, using
- generalized (u, u+v)-construction [i] based on
(125−9, 125, large)-Net in Base 8 — Upper bound on s
There is no (116, 125, large)-net in base 8, because
- 7 times m-reduction [i] would yield (116, 118, large)-net in base 8, but