Best Known (209−6, 209, s)-Nets in Base 3
(209−6, 209, large)-Net over F3 — Constructive and digital
Digital (203, 209, large)-net over F3, using
- 310 times duplication [i] based on digital (193, 199, large)-net over F3, using
- t-expansion [i] based on digital (188, 199, large)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (41, 46, 4194301)-net over F3, using
- net defined by OOA [i] based on linear OOA(346, 4194301, F3, 5, 5) (dual of [(4194301, 5), 20971459, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(346, large, F3, 5) (dual of [large, large−46, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [i]
- OOA 2-folding and stacking with additional row [i] based on linear OA(346, large, F3, 5) (dual of [large, large−46, 6]-code), using
- net defined by OOA [i] based on linear OOA(346, 4194301, F3, 5, 5) (dual of [(4194301, 5), 20971459, 6]-NRT-code), using
- digital (142, 153, 5033160)-net over F3, using
- trace code for nets [i] based on digital (40, 51, 1677720)-net over F27, using
- net defined by OOA [i] based on linear OOA(2751, 1677720, F27, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(2751, 8388601, F27, 11) (dual of [8388601, 8388550, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(2751, large, F27, 11) (dual of [large, large−51, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2751, large, F27, 11) (dual of [large, large−51, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(2751, 8388601, F27, 11) (dual of [8388601, 8388550, 12]-code), using
- net defined by OOA [i] based on linear OOA(2751, 1677720, F27, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
- trace code for nets [i] based on digital (40, 51, 1677720)-net over F27, using
- digital (41, 46, 4194301)-net over F3, using
- (u, u+v)-construction [i] based on
- t-expansion [i] based on digital (188, 199, large)-net over F3, using
(209−6, 209, large)-Net in Base 3 — Upper bound on s
There is no (203, 209, large)-net in base 3, because
- 4 times m-reduction [i] would yield (203, 205, large)-net in base 3, but