Best Known (75, 75+9, s)-Nets in Base 8
(75, 75+9, 4210695)-Net over F8 — Constructive and digital
Digital (75, 84, 4210695)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (14, 18, 16395)-net over F8, using
- net defined by OOA [i] based on linear OOA(818, 16395, F8, 4, 4) (dual of [(16395, 4), 65562, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(818, 16395, F8, 3, 4) (dual of [(16395, 3), 49167, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(82, 9, F8, 3, 2) (dual of [(9, 3), 25, 3]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(3;25,8) [i]
- linear OOA(816, 16386, F8, 3, 4) (dual of [(16386, 3), 49142, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(816, 32772, F8, 4) (dual of [32772, 32756, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(816, 32773, F8, 4) (dual of [32773, 32757, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(816, 32768, F8, 4) (dual of [32768, 32752, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(811, 32768, F8, 3) (dual of [32768, 32757, 4]-code or 32768-cap in PG(10,8)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(80, 5, F8, 0) (dual of [5, 5, 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
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- discarding factors / shortening the dual code based on linear OA(816, 32773, F8, 4) (dual of [32773, 32757, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(816, 32772, F8, 4) (dual of [32772, 32756, 5]-code), using
- linear OOA(82, 9, F8, 3, 2) (dual of [(9, 3), 25, 3]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(818, 16395, F8, 3, 4) (dual of [(16395, 3), 49167, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(818, 16395, F8, 4, 4) (dual of [(16395, 4), 65562, 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 (14, 18, 16395)-net over F8, using
(75, 75+9, 4259589)-Net in Base 8 — Constructive
(75, 84, 4259589)-net in base 8, using
- (u, u+v)-construction [i] based on
- (14, 18, 65289)-net in base 8, using
- net defined by OOA [i] based on OOA(818, 65289, S8, 4, 4), using
- appending kth column [i] based on OOA(818, 65289, S8, 3, 4), using
- (u, u+v)-construction [i] based on
- linear OOA(82, 9, F8, 3, 2) (dual of [(9, 3), 25, 3]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(3;25,8) [i]
- OOA(816, 65280, S8, 3, 4), using
- OA 2-folding and stacking [i] based on OA(816, 130560, S8, 4), using
- discarding parts of the base [i] based on linear OA(1612, 130560, F16, 4) (dual of [130560, 130548, 5]-code), using
- trace code [i] based on linear OA(2566, 65280, F256, 4) (dual of [65280, 65274, 5]-code), using
- 1 times truncation [i] based on linear OA(2567, 65281, F256, 5) (dual of [65281, 65274, 6]-code), using
- trace code [i] based on linear OA(2566, 65280, F256, 4) (dual of [65280, 65274, 5]-code), using
- discarding parts of the base [i] based on linear OA(1612, 130560, F16, 4) (dual of [130560, 130548, 5]-code), using
- OA 2-folding and stacking [i] based on OA(816, 130560, S8, 4), using
- linear OOA(82, 9, F8, 3, 2) (dual of [(9, 3), 25, 3]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on OOA(818, 65289, S8, 3, 4), using
- net defined by OOA [i] based on OOA(818, 65289, S8, 4, 4), 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
- (14, 18, 65289)-net in base 8, using
(75, 75+9, large)-Net over F8 — Digital
Digital (75, 84, large)-net over F8, using
- 83 times duplication [i] based on digital (72, 81, large)-net over F8, using
- t-expansion [i] based on digital (69, 81, large)-net over F8, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(881, large, F8, 12) (dual of [large, large−81, 13]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 88−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(881, large, F8, 12) (dual of [large, large−81, 13]-code), using
- t-expansion [i] based on digital (69, 81, large)-net over F8, using
(75, 75+9, large)-Net in Base 8 — Upper bound on s
There is no (75, 84, large)-net in base 8, because
- 7 times m-reduction [i] would yield (75, 77, large)-net in base 8, but