Best Known (34−10, 34, s)-Nets in Base 8
(34−10, 34, 820)-Net over F8 — Constructive and digital
Digital (24, 34, 820)-net over F8, using
- 81 times duplication [i] based on digital (23, 33, 820)-net over F8, using
- net defined by OOA [i] based on linear OOA(833, 820, F8, 10, 10) (dual of [(820, 10), 8167, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(833, 4100, F8, 10) (dual of [4100, 4067, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(833, 4096, F8, 10) (dual of [4096, 4063, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(829, 4096, F8, 9) (dual of [4096, 4067, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(80, 4, F8, 0) (dual of [4, 4, 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
- OA 5-folding and stacking [i] based on linear OA(833, 4100, F8, 10) (dual of [4100, 4067, 11]-code), using
- net defined by OOA [i] based on linear OOA(833, 820, F8, 10, 10) (dual of [(820, 10), 8167, 11]-NRT-code), using
(34−10, 34, 2852)-Net over F8 — Digital
Digital (24, 34, 2852)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(834, 2852, F8, 10) (dual of [2852, 2818, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(834, 4101, F8, 10) (dual of [4101, 4067, 11]-code), using
- 1 times code embedding in larger space [i] based on linear OA(833, 4100, F8, 10) (dual of [4100, 4067, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(833, 4096, F8, 10) (dual of [4096, 4063, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(829, 4096, F8, 9) (dual of [4096, 4067, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(80, 4, F8, 0) (dual of [4, 4, 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(833, 4100, F8, 10) (dual of [4100, 4067, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(834, 4101, F8, 10) (dual of [4101, 4067, 11]-code), using
(34−10, 34, 514929)-Net in Base 8 — Upper bound on s
There is no (24, 34, 514930)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 5 070623 603779 189070 713454 255628 > 834 [i]