Best Known (85, 85+10, s)-Nets in Base 8
(85, 85+10, 3371826)-Net over F8 — Constructive and digital
Digital (85, 95, 3371826)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (16, 21, 16386)-net over F8, using
- net defined by OOA [i] based on linear OOA(821, 16386, F8, 5, 5) (dual of [(16386, 5), 81909, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(821, 32773, F8, 5) (dual of [32773, 32752, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(821, 32768, F8, 5) (dual of [32768, 32747, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- 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(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(4) ⊂ Ce(3) [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OA(821, 32773, F8, 5) (dual of [32773, 32752, 6]-code), using
- net defined by OOA [i] based on linear OOA(821, 16386, F8, 5, 5) (dual of [(16386, 5), 81909, 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 (16, 21, 16386)-net over F8, using
(85, 85+10, 3420729)-Net in Base 8 — Constructive
(85, 95, 3420729)-net in base 8, using
- (u, u+v)-construction [i] based on
- (16, 21, 65289)-net in base 8, using
- net defined by OOA [i] based on OOA(821, 65289, S8, 6, 5), using
- OOA stacking with additional row [i] based on OOA(821, 65290, S8, 2, 5), using
- (u, u+v)-construction [i] based on
- linear OOA(82, 9, F8, 2, 2) (dual of [(9, 2), 16, 3]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(2;16,8) [i]
- 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(82, 9, F8, 2, 2) (dual of [(9, 2), 16, 3]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on OOA(821, 65290, S8, 2, 5), using
- net defined by OOA [i] based on OOA(821, 65289, 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
- (16, 21, 65289)-net in base 8, using
(85, 85+10, large)-Net over F8 — Digital
Digital (85, 95, large)-net over F8, using
- t-expansion [i] based on digital (83, 95, large)-net over F8, using
- 2 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
- 2 times m-reduction [i] based on digital (83, 97, large)-net over F8, using
(85, 85+10, large)-Net in Base 8 — Upper bound on s
There is no (85, 95, large)-net in base 8, because
- 8 times m-reduction [i] would yield (85, 87, large)-net in base 8, but