Best Known (78−10, 78, s)-Nets in Base 3
(78−10, 78, 106294)-Net over F3 — Constructive and digital
Digital (68, 78, 106294)-net over F3, using
- net defined by OOA [i] based on linear OOA(378, 106294, F3, 10, 10) (dual of [(106294, 10), 1062862, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(378, 531470, F3, 10) (dual of [531470, 531392, 11]-code), using
- 1 times code embedding in larger space [i] based on linear OA(377, 531469, F3, 10) (dual of [531469, 531392, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- linear OA(373, 531441, F3, 10) (dual of [531441, 531368, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(349, 531441, F3, 7) (dual of [531441, 531392, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(34, 28, F3, 2) (dual of [28, 24, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(377, 531469, F3, 10) (dual of [531469, 531392, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(378, 531470, F3, 10) (dual of [531470, 531392, 11]-code), using
(78−10, 78, 255889)-Net over F3 — Digital
Digital (68, 78, 255889)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(378, 255889, F3, 2, 10) (dual of [(255889, 2), 511700, 11]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(378, 265735, F3, 2, 10) (dual of [(265735, 2), 531392, 11]-NRT-code), using
- OOA 2-folding [i] based on linear OA(378, 531470, F3, 10) (dual of [531470, 531392, 11]-code), using
- 1 times code embedding in larger space [i] based on linear OA(377, 531469, F3, 10) (dual of [531469, 531392, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- linear OA(373, 531441, F3, 10) (dual of [531441, 531368, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(349, 531441, F3, 7) (dual of [531441, 531392, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(34, 28, F3, 2) (dual of [28, 24, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(377, 531469, F3, 10) (dual of [531469, 531392, 11]-code), using
- OOA 2-folding [i] based on linear OA(378, 531470, F3, 10) (dual of [531470, 531392, 11]-code), using
- discarding factors / shortening the dual code based on linear OOA(378, 265735, F3, 2, 10) (dual of [(265735, 2), 531392, 11]-NRT-code), using
(78−10, 78, large)-Net in Base 3 — Upper bound on s
There is no (68, 78, large)-net in base 3, because
- 8 times m-reduction [i] would yield (68, 70, large)-net in base 3, but