Best Known (169, 186, s)-Nets in Base 3
(169, 186, 1048615)-Net over F3 — Constructive and digital
Digital (169, 186, 1048615)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (12, 20, 40)-net over F3, using
- trace code for nets [i] based on digital (2, 10, 20)-net over F9, using
- net from sequence [i] based on digital (2, 19)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 2 and N(F) ≥ 20, using
- net from sequence [i] based on digital (2, 19)-sequence over F9, using
- trace code for nets [i] based on digital (2, 10, 20)-net over F9, using
- digital (149, 166, 1048575)-net over F3, using
- net defined by OOA [i] based on linear OOA(3166, 1048575, F3, 17, 17) (dual of [(1048575, 17), 17825609, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3166, 8388601, F3, 17) (dual of [8388601, 8388435, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(3166, large, F3, 17) (dual of [large, large−166, 18]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(3166, large, F3, 17) (dual of [large, large−166, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3166, 8388601, F3, 17) (dual of [8388601, 8388435, 18]-code), using
- net defined by OOA [i] based on linear OOA(3166, 1048575, F3, 17, 17) (dual of [(1048575, 17), 17825609, 18]-NRT-code), using
- digital (12, 20, 40)-net over F3, using
(169, 186, 4194346)-Net over F3 — Digital
Digital (169, 186, 4194346)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3186, 4194346, F3, 2, 17) (dual of [(4194346, 2), 8388506, 18]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(320, 45, F3, 2, 8) (dual of [(45, 2), 70, 9]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(320, 45, F3, 8) (dual of [45, 25, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(320, 46, F3, 8) (dual of [46, 26, 9]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- linear OA(33, 11, F3, 2) (dual of [11, 8, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- discarding factors / shortening the dual code based on linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- linear OA(35, 11, F3, 4) (dual of [11, 6, 5]-code), using
- Golay code G(3) [i]
- linear OA(312, 24, F3, 8) (dual of [24, 12, 9]-code), using
- extended quadratic residue code Qe(24,3) [i]
- linear OA(33, 11, F3, 2) (dual of [11, 8, 3]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(320, 46, F3, 8) (dual of [46, 26, 9]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(320, 45, F3, 8) (dual of [45, 25, 9]-code), using
- linear OOA(3166, 4194301, F3, 2, 17) (dual of [(4194301, 2), 8388436, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3166, 8388602, F3, 17) (dual of [8388602, 8388436, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(3166, large, F3, 17) (dual of [large, large−166, 18]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(3166, large, F3, 17) (dual of [large, large−166, 18]-code), using
- OOA 2-folding [i] based on linear OA(3166, 8388602, F3, 17) (dual of [8388602, 8388436, 18]-code), using
- linear OOA(320, 45, F3, 2, 8) (dual of [(45, 2), 70, 9]-NRT-code), using
- (u, u+v)-construction [i] based on
(169, 186, large)-Net in Base 3 — Upper bound on s
There is no (169, 186, large)-net in base 3, because
- 15 times m-reduction [i] would yield (169, 171, large)-net in base 3, but