Best Known (82, 82+10, s)-Nets in Base 8
(82, 82+10, 3360057)-Net over F8 — Constructive and digital
Digital (82, 92, 3360057)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (13, 18, 4617)-net over F8, using
- net defined by OOA [i] based on linear OOA(818, 4617, F8, 6, 5) (dual of [(4617, 6), 27684, 6]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(818, 4618, F8, 2, 5) (dual of [(4618, 2), 9218, 6]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(84, 585, F8, 2, 2) (dual of [(585, 2), 1166, 3]-NRT-code), using
- appending kth column [i] based on linear OA(84, 585, F8, 2) (dual of [585, 581, 3]-code), using
- Hamming code H(4,8) [i]
- appending kth column [i] based on linear OA(84, 585, F8, 2) (dual of [585, 581, 3]-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(84, 585, F8, 2, 2) (dual of [(585, 2), 1166, 3]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(818, 4618, F8, 2, 5) (dual of [(4618, 2), 9218, 6]-NRT-code), using
- net defined by OOA [i] based on linear OOA(818, 4617, F8, 6, 5) (dual of [(4617, 6), 27684, 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 (13, 18, 4617)-net over F8, using
(82, 82+10, 3388080)-Net in Base 8 — Constructive
(82, 92, 3388080)-net in base 8, using
- base change [i] based on digital (59, 69, 3388080)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (8, 13, 32640)-net over F16, using
- net defined by OOA [i] based on linear OOA(1613, 32640, F16, 5, 5) (dual of [(32640, 5), 163187, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(1613, 65281, F16, 5) (dual of [65281, 65268, 6]-code), using
- net defined by OOA [i] based on linear OOA(1613, 32640, F16, 5, 5) (dual of [(32640, 5), 163187, 6]-NRT-code), using
- digital (46, 56, 3355440)-net over F16, using
- trace code for nets [i] based on digital (18, 28, 1677720)-net over F256, using
- net defined by OOA [i] based on linear OOA(25628, 1677720, F256, 10, 10) (dual of [(1677720, 10), 16777172, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(25628, 8388600, F256, 10) (dual of [8388600, 8388572, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(25628, large, F256, 10) (dual of [large, large−28, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(25628, large, F256, 10) (dual of [large, large−28, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(25628, 8388600, F256, 10) (dual of [8388600, 8388572, 11]-code), using
- net defined by OOA [i] based on linear OOA(25628, 1677720, F256, 10, 10) (dual of [(1677720, 10), 16777172, 11]-NRT-code), using
- trace code for nets [i] based on digital (18, 28, 1677720)-net over F256, using
- digital (8, 13, 32640)-net over F16, using
- (u, u+v)-construction [i] based on
(82, 82+10, large)-Net over F8 — Digital
Digital (82, 92, large)-net over F8, using
- 83 times duplication [i] based on digital (79, 89, large)-net over F8, using
- t-expansion [i] based on digital (76, 89, large)-net over F8, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(889, large, F8, 13) (dual of [large, large−89, 14]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 88−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(889, large, F8, 13) (dual of [large, large−89, 14]-code), using
- t-expansion [i] based on digital (76, 89, large)-net over F8, using
(82, 82+10, large)-Net in Base 8 — Upper bound on s
There is no (82, 92, large)-net in base 8, because
- 8 times m-reduction [i] would yield (82, 84, large)-net in base 8, but