Best Known (73, 73+8, s)-Nets in Base 3
(73, 73+8, 2097158)-Net over F3 — Constructive and digital
Digital (73, 81, 2097158)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 5, 8)-net over F3, using
- 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
(73, 73+8, 4194309)-Net over F3 — Digital
Digital (73, 81, 4194309)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(381, 4194309, F3, 2, 8) (dual of [(4194309, 2), 8388537, 9]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(35, 8, F3, 2, 4) (dual of [(8, 2), 11, 5]-NRT-code), using
- extracting embedded OOA [i] based on digital (1, 5, 8)-net over F3, using
- 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
- linear OOA(35, 8, F3, 2, 4) (dual of [(8, 2), 11, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
(73, 73+8, large)-Net in Base 3 — Upper bound on s
There is no (73, 81, large)-net in base 3, because
- 6 times m-reduction [i] would yield (73, 75, large)-net in base 3, but