Best Known (68−9, 68, s)-Nets in Base 8
(68−9, 68, 4194301)-Net over F8 — Constructive and digital
Digital (59, 68, 4194301)-net over F8, using
- net defined by OOA [i] based on linear OOA(868, 4194301, F8, 10, 9) (dual of [(4194301, 10), 41942942, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(868, large, F8, 2, 9), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(866, 8388602, F8, 2, 9) (dual of [(8388602, 2), 16777138, 10]-NRT-code), using
- trace code [i] based on linear OOA(6433, 4194301, F64, 2, 9) (dual of [(4194301, 2), 8388569, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6433, 8388602, F64, 9) (dual of [8388602, 8388569, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(6433, large, F64, 9) (dual of [large, large−33, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(6433, large, F64, 9) (dual of [large, large−33, 10]-code), using
- OOA 2-folding [i] based on linear OA(6433, 8388602, F64, 9) (dual of [8388602, 8388569, 10]-code), using
- trace code [i] based on linear OOA(6433, 4194301, F64, 2, 9) (dual of [(4194301, 2), 8388569, 10]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(866, 8388602, F8, 2, 9) (dual of [(8388602, 2), 16777138, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(868, large, F8, 2, 9), using
(68−9, 68, large)-Net over F8 — Digital
Digital (59, 68, large)-net over F8, using
- 83 times duplication [i] based on digital (56, 65, large)-net over F8, using
- t-expansion [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
- t-expansion [i] based on digital (55, 65, large)-net over F8, using
(68−9, 68, large)-Net in Base 8 — Upper bound on s
There is no (59, 68, large)-net in base 8, because
- 7 times m-reduction [i] would yield (59, 61, large)-net in base 8, but