Best Known (42−10, 42, s)-Nets in Base 8
(42−10, 42, 6554)-Net over F8 — Constructive and digital
Digital (32, 42, 6554)-net over F8, using
- 81 times duplication [i] based on digital (31, 41, 6554)-net over F8, using
- net defined by OOA [i] based on linear OOA(841, 6554, F8, 10, 10) (dual of [(6554, 10), 65499, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(841, 32770, F8, 10) (dual of [32770, 32729, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(841, 32773, F8, 10) (dual of [32773, 32732, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(841, 32768, F8, 10) (dual of [32768, 32727, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(836, 32768, F8, 9) (dual of [32768, 32732, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(9) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(841, 32773, F8, 10) (dual of [32773, 32732, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(841, 32770, F8, 10) (dual of [32770, 32729, 11]-code), using
- net defined by OOA [i] based on linear OOA(841, 6554, F8, 10, 10) (dual of [(6554, 10), 65499, 11]-NRT-code), using
(42−10, 42, 22848)-Net over F8 — Digital
Digital (32, 42, 22848)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(842, 22848, F8, 10) (dual of [22848, 22806, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(842, 32774, F8, 10) (dual of [32774, 32732, 11]-code), using
- 1 times code embedding in larger space [i] based on linear OA(841, 32773, F8, 10) (dual of [32773, 32732, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(841, 32768, F8, 10) (dual of [32768, 32727, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(836, 32768, F8, 9) (dual of [32768, 32732, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(9) ⊂ Ce(8) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(841, 32773, F8, 10) (dual of [32773, 32732, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(842, 32774, F8, 10) (dual of [32774, 32732, 11]-code), using
(42−10, 42, large)-Net in Base 8 — Upper bound on s
There is no (32, 42, large)-net in base 8, because
- 8 times m-reduction [i] would yield (32, 34, large)-net in base 8, but