Best Known (137−14, 137, s)-Nets in Base 8
(137−14, 137, 2407666)-Net over F8 — Constructive and digital
Digital (123, 137, 2407666)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (24, 31, 10924)-net over F8, using
- net defined by OOA [i] based on linear OOA(831, 10924, F8, 7, 7) (dual of [(10924, 7), 76437, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(831, 32773, F8, 7) (dual of [32773, 32742, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(831, 32768, F8, 7) (dual of [32768, 32737, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(826, 32768, F8, 6) (dual of [32768, 32742, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(6) ⊂ Ce(5) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(831, 32773, F8, 7) (dual of [32773, 32742, 8]-code), using
- net defined by OOA [i] based on linear OOA(831, 10924, F8, 7, 7) (dual of [(10924, 7), 76437, 8]-NRT-code), using
- digital (92, 106, 2396742)-net over F8, using
- trace code for nets [i] based on digital (39, 53, 1198371)-net over F64, using
- net defined by OOA [i] based on linear OOA(6453, 1198371, F64, 14, 14) (dual of [(1198371, 14), 16777141, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(6453, 8388597, F64, 14) (dual of [8388597, 8388544, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(6453, large, F64, 14) (dual of [large, large−53, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(6453, large, F64, 14) (dual of [large, large−53, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(6453, 8388597, F64, 14) (dual of [8388597, 8388544, 15]-code), using
- net defined by OOA [i] based on linear OOA(6453, 1198371, F64, 14, 14) (dual of [(1198371, 14), 16777141, 15]-NRT-code), using
- trace code for nets [i] based on digital (39, 53, 1198371)-net over F64, using
- digital (24, 31, 10924)-net over F8, using
(137−14, 137, large)-Net over F8 — Digital
Digital (123, 137, large)-net over F8, using
- t-expansion [i] based on digital (119, 137, large)-net over F8, using
- 2 times m-reduction [i] based on digital (119, 139, large)-net over F8, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8139, large, F8, 20) (dual of [large, large−139, 21]-code), using
- 2 times code embedding in larger space [i] based on linear OA(8137, large, F8, 20) (dual of [large, large−137, 21]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 88−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [i]
- 2 times code embedding in larger space [i] based on linear OA(8137, large, F8, 20) (dual of [large, large−137, 21]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8139, large, F8, 20) (dual of [large, large−139, 21]-code), using
- 2 times m-reduction [i] based on digital (119, 139, large)-net over F8, using
(137−14, 137, large)-Net in Base 8 — Upper bound on s
There is no (123, 137, large)-net in base 8, because
- 12 times m-reduction [i] would yield (123, 125, large)-net in base 8, but