Best Known (78−8, 78, s)-Nets in Base 3
(78−8, 78, 2097150)-Net over F3 — Constructive and digital
Digital (70, 78, 2097150)-net over F3, using
- 32 times duplication [i] based on digital (68, 76, 2097150)-net over F3, using
- net defined by OOA [i] based on linear OOA(376, 2097150, F3, 8, 8) (dual of [(2097150, 8), 16777124, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(376, 8388600, F3, 8) (dual of [8388600, 8388524, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(376, large, F3, 8) (dual of [large, large−76, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(376, large, F3, 8) (dual of [large, large−76, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(376, 8388600, F3, 8) (dual of [8388600, 8388524, 9]-code), using
- net defined by OOA [i] based on linear OOA(376, 2097150, F3, 8, 8) (dual of [(2097150, 8), 16777124, 9]-NRT-code), using
(78−8, 78, 4194302)-Net over F3 — Digital
Digital (70, 78, 4194302)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(378, 4194302, F3, 2, 8) (dual of [(4194302, 2), 8388526, 9]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(376, 4194301, F3, 2, 8) (dual of [(4194301, 2), 8388526, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(376, 8388602, F3, 8) (dual of [8388602, 8388526, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(376, large, F3, 8) (dual of [large, large−76, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(376, large, F3, 8) (dual of [large, large−76, 9]-code), using
- OOA 2-folding [i] based on linear OA(376, 8388602, F3, 8) (dual of [8388602, 8388526, 9]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(376, 4194301, F3, 2, 8) (dual of [(4194301, 2), 8388526, 9]-NRT-code), using
(78−8, 78, large)-Net in Base 3 — Upper bound on s
There is no (70, 78, large)-net in base 3, because
- 6 times m-reduction [i] would yield (70, 72, large)-net in base 3, but