Best Known (209−15, 209, s)-Nets in Base 3
(209−15, 209, 2396742)-Net over F3 — Constructive and digital
Digital (194, 209, 2396742)-net over F3, using
- 32 times duplication [i] based on digital (192, 207, 2396742)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (50, 57, 1594323)-net over F3, using
- net defined by OOA [i] based on linear OOA(357, 1594323, F3, 7, 7) (dual of [(1594323, 7), 11160204, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(357, 4782970, F3, 7) (dual of [4782970, 4782913, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4782970 | 328−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- OOA 3-folding and stacking with additional row [i] based on linear OA(357, 4782970, F3, 7) (dual of [4782970, 4782913, 8]-code), using
- net defined by OOA [i] based on linear OOA(357, 1594323, F3, 7, 7) (dual of [(1594323, 7), 11160204, 8]-NRT-code), using
- digital (135, 150, 1198371)-net over F3, using
- net defined by OOA [i] based on linear OOA(3150, 1198371, F3, 15, 15) (dual of [(1198371, 15), 17975415, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(3150, 8388598, F3, 15) (dual of [8388598, 8388448, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(3150, large, F3, 15) (dual of [large, large−150, 16]-code), using
- the primitive narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(3150, large, F3, 15) (dual of [large, large−150, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(3150, 8388598, F3, 15) (dual of [8388598, 8388448, 16]-code), using
- net defined by OOA [i] based on linear OOA(3150, 1198371, F3, 15, 15) (dual of [(1198371, 15), 17975415, 16]-NRT-code), using
- digital (50, 57, 1594323)-net over F3, using
- (u, u+v)-construction [i] based on
(209−15, 209, large)-Net over F3 — Digital
Digital (194, 209, large)-net over F3, using
- 36 times duplication [i] based on digital (188, 203, large)-net over F3, using
- t-expansion [i] based on digital (186, 203, large)-net over F3, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3203, large, F3, 17) (dual of [large, large−203, 18]-code), using
- 37 times code embedding in larger space [i] 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]
- 37 times code embedding in larger space [i] based on linear OA(3166, large, F3, 17) (dual of [large, large−166, 18]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3203, large, F3, 17) (dual of [large, large−203, 18]-code), using
- t-expansion [i] based on digital (186, 203, large)-net over F3, using
(209−15, 209, large)-Net in Base 3 — Upper bound on s
There is no (194, 209, large)-net in base 3, because
- 13 times m-reduction [i] would yield (194, 196, large)-net in base 3, but