Best Known (109, 109+7, s)-Nets in Base 8
(109, 109+7, large)-Net over F8 — Constructive and digital
Digital (109, 116, large)-net over F8, using
- 3 times m-reduction [i] based on digital (109, 119, 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 (64, 74, 3355440)-net over F8, using
- trace code for nets [i] based on digital (27, 37, 1677720)-net over F64, using
- net defined by OOA [i] based on linear OOA(6437, 1677720, F64, 10, 10) (dual of [(1677720, 10), 16777163, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(6437, 8388600, F64, 10) (dual of [8388600, 8388563, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(6437, large, F64, 10) (dual of [large, large−37, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(6437, large, F64, 10) (dual of [large, large−37, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(6437, 8388600, F64, 10) (dual of [8388600, 8388563, 11]-code), using
- net defined by OOA [i] based on linear OOA(6437, 1677720, F64, 10, 10) (dual of [(1677720, 10), 16777163, 11]-NRT-code), using
- trace code for nets [i] based on digital (27, 37, 1677720)-net over F64, using
- digital (9, 12, 2931457)-net over F8, using
- generalized (u, u+v)-construction [i] based on
(109, 109+7, large)-Net in Base 8 — Upper bound on s
There is no (109, 116, large)-net in base 8, because
- 5 times m-reduction [i] would yield (109, 111, large)-net in base 8, but