Best Known (135, 135+12, s)-Nets in Base 8
(135, 135+12, 5608786)-Net over F8 — Constructive and digital
Digital (135, 147, 5608786)-net over F8, using
- generalized (u, u+v)-construction [i] based on
- digital (12, 16, 16386)-net over F8, using
- net defined by OOA [i] based on linear OOA(816, 16386, F8, 4, 4) (dual of [(16386, 4), 65528, 5]-NRT-code), using
- appending kth column [i] based on 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
- appending kth column [i] based on linear OOA(816, 16386, F8, 3, 4) (dual of [(16386, 3), 49142, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(816, 16386, F8, 4, 4) (dual of [(16386, 4), 65528, 5]-NRT-code), using
- digital (35, 41, 2796200)-net over F8, using
- s-reduction based on digital (35, 41, 2796201)-net over F8, using
- net defined by OOA [i] based on linear OOA(841, 2796201, F8, 6, 6) (dual of [(2796201, 6), 16777165, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(841, large, F8, 6) (dual of [large, large−41, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 88−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- OA 3-folding and stacking [i] based on linear OA(841, large, F8, 6) (dual of [large, large−41, 7]-code), using
- net defined by OOA [i] based on linear OOA(841, 2796201, F8, 6, 6) (dual of [(2796201, 6), 16777165, 7]-NRT-code), using
- s-reduction based on digital (35, 41, 2796201)-net over F8, using
- digital (78, 90, 2796200)-net over F8, using
- net defined by OOA [i] based on linear OOA(890, 2796200, F8, 14, 12) (dual of [(2796200, 14), 39146710, 13]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(890, 8388601, F8, 2, 12) (dual of [(8388601, 2), 16777112, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(890, 8388602, F8, 2, 12) (dual of [(8388602, 2), 16777114, 13]-NRT-code), using
- trace code [i] based on linear OOA(6445, 4194301, F64, 2, 12) (dual of [(4194301, 2), 8388557, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6445, 8388602, F64, 12) (dual of [8388602, 8388557, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(6445, large, F64, 12) (dual of [large, large−45, 13]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(6445, large, F64, 12) (dual of [large, large−45, 13]-code), using
- OOA 2-folding [i] based on linear OA(6445, 8388602, F64, 12) (dual of [8388602, 8388557, 13]-code), using
- trace code [i] based on linear OOA(6445, 4194301, F64, 2, 12) (dual of [(4194301, 2), 8388557, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(890, 8388602, F8, 2, 12) (dual of [(8388602, 2), 16777114, 13]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(890, 8388601, F8, 2, 12) (dual of [(8388601, 2), 16777112, 13]-NRT-code), using
- net defined by OOA [i] based on linear OOA(890, 2796200, F8, 14, 12) (dual of [(2796200, 14), 39146710, 13]-NRT-code), using
- digital (12, 16, 16386)-net over F8, using
(135, 135+12, large)-Net over F8 — Digital
Digital (135, 147, large)-net over F8, using
- t-expansion [i] based on digital (131, 147, large)-net over F8, using
- 6 times m-reduction [i] based on digital (131, 153, large)-net over F8, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8153, large, F8, 22) (dual of [large, large−153, 23]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 88−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8153, large, F8, 22) (dual of [large, large−153, 23]-code), using
- 6 times m-reduction [i] based on digital (131, 153, large)-net over F8, using
(135, 135+12, large)-Net in Base 8 — Upper bound on s
There is no (135, 147, large)-net in base 8, because
- 10 times m-reduction [i] would yield (135, 137, large)-net in base 8, but