Best Known (56, 56+10, s)-Nets in Base 8
(56, 56+10, 1677720)-Net over F8 — Constructive and digital
Digital (56, 66, 1677720)-net over F8, using
- 81 times duplication [i] based on digital (55, 65, 1677720)-net over F8, using
- net defined by OOA [i] based on linear OOA(865, 1677720, F8, 10, 10) (dual of [(1677720, 10), 16777135, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(865, 8388600, F8, 10) (dual of [8388600, 8388535, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(865, large, F8, 10) (dual of [large, large−65, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 88−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(865, large, F8, 10) (dual of [large, large−65, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(865, 8388600, F8, 10) (dual of [8388600, 8388535, 11]-code), using
- net defined by OOA [i] based on linear OOA(865, 1677720, F8, 10, 10) (dual of [(1677720, 10), 16777135, 11]-NRT-code), using
(56, 56+10, large)-Net over F8 — Digital
Digital (56, 66, large)-net over F8, using
- 81 times duplication [i] based on digital (55, 65, large)-net over F8, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(865, large, F8, 10) (dual of [large, large−65, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 88−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(865, large, F8, 10) (dual of [large, large−65, 11]-code), using
(56, 56+10, large)-Net in Base 8 — Upper bound on s
There is no (56, 66, large)-net in base 8, because
- 8 times m-reduction [i] would yield (56, 58, large)-net in base 8, but