Best Known (86, 86+10, s)-Nets in Base 8
(86, 86+10, 3387703)-Net over F8 — Constructive and digital
Digital (86, 96, 3387703)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (17, 22, 32263)-net over F8, using
- net defined by OOA [i] based on linear OOA(822, 32263, F8, 6, 5) (dual of [(32263, 6), 193556, 6]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(822, 32264, F8, 2, 5) (dual of [(32264, 2), 64506, 6]-NRT-code), using
- generalized (u, u+v)-construction [i] based on
- linear OOA(80, 4033, F8, 2, 0) (dual of [(4033, 2), 8066, 1]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(80, s, F8, 2, 0) with arbitrarily large s, using
- linear OOA(80, 4033, F8, 2, 0) (dual of [(4033, 2), 8066, 1]-NRT-code) (see above)
- linear OOA(80, 4033, F8, 2, 0) (dual of [(4033, 2), 8066, 1]-NRT-code) (see above)
- linear OOA(81, 4033, F8, 2, 1) (dual of [(4033, 2), 8065, 2]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(81, s, F8, 2, 1) with arbitrarily large s, using
- appending 1 arbitrary column [i] based on linear OA(81, s, F8, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- discarding factors / shortening the dual code based on linear OOA(81, s, F8, 2, 1) with arbitrarily large s, using
- linear OOA(81, 4033, F8, 2, 1) (dual of [(4033, 2), 8065, 2]-NRT-code) (see above)
- linear OOA(81, 4033, F8, 2, 1) (dual of [(4033, 2), 8065, 2]-NRT-code) (see above)
- linear OOA(85, 4033, F8, 2, 2) (dual of [(4033, 2), 8061, 3]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(85, 4681, F8, 2, 2) (dual of [(4681, 2), 9357, 3]-NRT-code), using
- appending kth column [i] based on linear OA(85, 4681, F8, 2) (dual of [4681, 4676, 3]-code), using
- Hamming code H(5,8) [i]
- appending kth column [i] based on linear OA(85, 4681, F8, 2) (dual of [4681, 4676, 3]-code), using
- discarding factors / shortening the dual code based on linear OOA(85, 4681, F8, 2, 2) (dual of [(4681, 2), 9357, 3]-NRT-code), using
- linear OOA(814, 4033, F8, 2, 5) (dual of [(4033, 2), 8052, 6]-NRT-code), using
- OOA 2-folding [i] based on linear OA(814, 8066, F8, 5) (dual of [8066, 8052, 6]-code), using
- trace code [i] based on linear OA(647, 4033, F64, 5) (dual of [4033, 4026, 6]-code), using
- OOA 2-folding [i] based on linear OA(814, 8066, F8, 5) (dual of [8066, 8052, 6]-code), using
- linear OOA(80, 4033, F8, 2, 0) (dual of [(4033, 2), 8066, 1]-NRT-code), using
- generalized (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(822, 32264, F8, 2, 5) (dual of [(32264, 2), 64506, 6]-NRT-code), using
- net defined by OOA [i] based on linear OOA(822, 32263, F8, 6, 5) (dual of [(32263, 6), 193556, 6]-NRT-code), 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 (17, 22, 32263)-net over F8, using
(86, 86+10, 3420793)-Net in Base 8 — Constructive
(86, 96, 3420793)-net in base 8, using
- (u, u+v)-construction [i] based on
- (17, 22, 65353)-net in base 8, using
- net defined by OOA [i] based on OOA(822, 65353, S8, 6, 5), using
- OOA stacking with additional row [i] based on OOA(822, 65354, S8, 2, 5), using
- (u, u+v)-construction [i] based on
- linear OOA(83, 73, F8, 2, 2) (dual of [(73, 2), 143, 3]-NRT-code), using
- appending kth column [i] based on linear OA(83, 73, F8, 2) (dual of [73, 70, 3]-code), using
- Hamming code H(3,8) [i]
- appending kth column [i] based on linear OA(83, 73, F8, 2) (dual of [73, 70, 3]-code), using
- OOA(819, 65281, S8, 2, 5), using
- OOA 2-folding [i] based on OA(819, 130562, S8, 5), using
- discarding parts of the base [i] based on linear OA(1614, 130562, F16, 5) (dual of [130562, 130548, 6]-code), using
- trace code [i] based on linear OA(2567, 65281, F256, 5) (dual of [65281, 65274, 6]-code), using
- discarding parts of the base [i] based on linear OA(1614, 130562, F16, 5) (dual of [130562, 130548, 6]-code), using
- OOA 2-folding [i] based on OA(819, 130562, S8, 5), using
- linear OOA(83, 73, F8, 2, 2) (dual of [(73, 2), 143, 3]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on OOA(822, 65354, S8, 2, 5), using
- net defined by OOA [i] based on OOA(822, 65353, S8, 6, 5), 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
- (17, 22, 65353)-net in base 8, using
(86, 86+10, large)-Net over F8 — Digital
Digital (86, 96, large)-net over F8, using
- t-expansion [i] based on digital (83, 96, large)-net over F8, using
- 1 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
- 1 times m-reduction [i] based on digital (83, 97, large)-net over F8, using
(86, 86+10, large)-Net in Base 8 — Upper bound on s
There is no (86, 96, large)-net in base 8, because
- 8 times m-reduction [i] would yield (86, 88, large)-net in base 8, but