Best Known (76−9, 76, s)-Nets in Base 8
(76−9, 76, 4194558)-Net over F8 — Constructive and digital
Digital (67, 76, 4194558)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (6, 10, 258)-net over F8, using
- net defined by OOA [i] based on linear OOA(810, 258, F8, 4, 4) (dual of [(258, 4), 1022, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(810, 258, F8, 3, 4) (dual of [(258, 3), 764, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(810, 516, F8, 4) (dual of [516, 506, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(810, 517, F8, 4) (dual of [517, 507, 5]-code), using
- construction XX applied to C1 = C([510,1]), C2 = C([0,2]), C3 = C1 + C2 = C([0,1]), and C∩ = C1 ∩ C2 = C([510,2]) [i] based on
- linear OA(87, 511, F8, 3) (dual of [511, 504, 4]-code or 511-cap in PG(6,8)), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {−1,0,1}, and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(87, 511, F8, 3) (dual of [511, 504, 4]-code or 511-cap in PG(6,8)), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(810, 511, F8, 4) (dual of [511, 501, 5]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {−1,0,1,2}, and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(84, 511, F8, 2) (dual of [511, 507, 3]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(80, 3, F8, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(80, 3, F8, 0) (dual of [3, 3, 1]-code) (see above)
- construction XX applied to C1 = C([510,1]), C2 = C([0,2]), C3 = C1 + C2 = C([0,1]), and C∩ = C1 ∩ C2 = C([510,2]) [i] based on
- discarding factors / shortening the dual code based on linear OA(810, 517, F8, 4) (dual of [517, 507, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(810, 516, F8, 4) (dual of [516, 506, 5]-code), using
- appending kth column [i] based on linear OOA(810, 258, F8, 3, 4) (dual of [(258, 3), 764, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(810, 258, F8, 4, 4) (dual of [(258, 4), 1022, 5]-NRT-code), using
- digital (57, 66, 4194300)-net over F8, using
- net defined by OOA [i] based on linear OOA(866, 4194300, F8, 10, 9) (dual of [(4194300, 10), 41942934, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(866, 8388601, F8, 2, 9) (dual of [(8388601, 2), 16777136, 10]-NRT-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code 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(866, 8388601, F8, 2, 9) (dual of [(8388601, 2), 16777136, 10]-NRT-code), using
- net defined by OOA [i] based on linear OOA(866, 4194300, F8, 10, 9) (dual of [(4194300, 10), 41942934, 10]-NRT-code), using
- digital (6, 10, 258)-net over F8, using
(76−9, 76, 4194796)-Net in Base 8 — Constructive
(67, 76, 4194796)-net in base 8, using
- (u, u+v)-construction [i] based on
- (6, 10, 496)-net in base 8, using
- base change [i] based on digital (2, 6, 496)-net over F32, using
- net defined by OOA [i] based on linear OOA(326, 496, F32, 4, 4) (dual of [(496, 4), 1978, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(326, 992, F32, 4) (dual of [992, 986, 5]-code), using
- 1 times truncation [i] based on linear OA(327, 993, F32, 5) (dual of [993, 986, 6]-code), using
- OA 2-folding and stacking [i] based on linear OA(326, 992, F32, 4) (dual of [992, 986, 5]-code), using
- net defined by OOA [i] based on linear OOA(326, 496, F32, 4, 4) (dual of [(496, 4), 1978, 5]-NRT-code), using
- base change [i] based on digital (2, 6, 496)-net over F32, using
- digital (57, 66, 4194300)-net over F8, using
- net defined by OOA [i] based on linear OOA(866, 4194300, F8, 10, 9) (dual of [(4194300, 10), 41942934, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(866, 8388601, F8, 2, 9) (dual of [(8388601, 2), 16777136, 10]-NRT-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code 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(866, 8388601, F8, 2, 9) (dual of [(8388601, 2), 16777136, 10]-NRT-code), using
- net defined by OOA [i] based on linear OOA(866, 4194300, F8, 10, 9) (dual of [(4194300, 10), 41942934, 10]-NRT-code), using
- (6, 10, 496)-net in base 8, using
(76−9, 76, large)-Net over F8 — Digital
Digital (67, 76, large)-net over F8, using
- 83 times duplication [i] based on digital (64, 73, large)-net over F8, using
- t-expansion [i] based on digital (62, 73, large)-net over F8, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(873, large, F8, 11) (dual of [large, large−73, 12]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 88−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(873, large, F8, 11) (dual of [large, large−73, 12]-code), using
- t-expansion [i] based on digital (62, 73, large)-net over F8, using
(76−9, 76, large)-Net in Base 8 — Upper bound on s
There is no (67, 76, large)-net in base 8, because
- 7 times m-reduction [i] would yield (67, 69, large)-net in base 8, but