Best Known (83, 83+9, s)-Nets in Base 8
(83, 83+9, 8388600)-Net over F8 — Constructive and digital
Digital (83, 92, 8388600)-net over F8, using
- trace code for nets [i] based on digital (37, 46, 4194300)-net over F64, using
- net defined by OOA [i] based on linear OOA(6446, 4194300, F64, 10, 9) (dual of [(4194300, 10), 41942954, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(6446, 8388601, F64, 2, 9) (dual of [(8388601, 2), 16777156, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6446, 8388602, F64, 2, 9) (dual of [(8388602, 2), 16777158, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(6413, 4194301, F64, 2, 4) (dual of [(4194301, 2), 8388589, 5]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6413, 8388602, F64, 4) (dual of [8388602, 8388589, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(6413, large, F64, 4) (dual of [large, large−13, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(6413, large, F64, 4) (dual of [large, large−13, 5]-code), using
- OOA 2-folding [i] based on linear OA(6413, 8388602, F64, 4) (dual of [8388602, 8388589, 5]-code), using
- 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
- linear OOA(6413, 4194301, F64, 2, 4) (dual of [(4194301, 2), 8388589, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OOA(6446, 8388602, F64, 2, 9) (dual of [(8388602, 2), 16777158, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(6446, 8388601, F64, 2, 9) (dual of [(8388601, 2), 16777156, 10]-NRT-code), using
- net defined by OOA [i] based on linear OOA(6446, 4194300, F64, 10, 9) (dual of [(4194300, 10), 41942954, 10]-NRT-code), using
(83, 83+9, large)-Net over F8 — Digital
Digital (83, 92, large)-net over F8, using
- 5 times m-reduction [i] based on digital (83, 97, large)-net over F8, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(897, large, F8, 14) (dual of [large, large−97, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 88−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(897, large, F8, 14) (dual of [large, large−97, 15]-code), using
(83, 83+9, large)-Net in Base 8 — Upper bound on s
There is no (83, 92, large)-net in base 8, because
- 7 times m-reduction [i] would yield (83, 85, large)-net in base 8, but