Best Known (56, 56+6, s)-Nets in Base 8
(56, 56+6, large)-Net over F8 — Constructive and digital
Digital (56, 62, large)-net over F8, using
- t-expansion [i] based on digital (55, 62, large)-net over F8, using
- (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 (43, 50, 5592401)-net over F8, using
- net defined by OOA [i] based on linear OOA(850, 5592401, F8, 9, 7) (dual of [(5592401, 9), 50331559, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(850, 5592402, F8, 3, 7) (dual of [(5592402, 3), 16777156, 8]-NRT-code), using
- trace code [i] based on linear OOA(6425, 2796201, F64, 3, 7) (dual of [(2796201, 3), 8388578, 8]-NRT-code), using
- OOA 3-folding [i] based on linear OA(6425, large, F64, 7) (dual of [large, large−25, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- OOA 3-folding [i] based on linear OA(6425, large, F64, 7) (dual of [large, large−25, 8]-code), using
- trace code [i] based on linear OOA(6425, 2796201, F64, 3, 7) (dual of [(2796201, 3), 8388578, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(850, 5592402, F8, 3, 7) (dual of [(5592402, 3), 16777156, 8]-NRT-code), using
- net defined by OOA [i] based on linear OOA(850, 5592401, F8, 9, 7) (dual of [(5592401, 9), 50331559, 8]-NRT-code), using
- digital (9, 12, 2931457)-net over F8, using
- (u, u+v)-construction [i] based on
(56, 56+6, large)-Net in Base 8 — Upper bound on s
There is no (56, 62, large)-net in base 8, because
- 4 times m-reduction [i] would yield (56, 58, large)-net in base 8, but