Best Known (207, 207+19, s)-Nets in Base 3
(207, 207+19, 1062882)-Net over F3 — Constructive and digital
Digital (207, 226, 1062882)-net over F3, using
- trace code for nets [i] based on digital (94, 113, 531441)-net over F9, using
- net defined by OOA [i] based on linear OOA(9113, 531441, F9, 19, 19) (dual of [(531441, 19), 10097266, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(9113, 4782970, F9, 19) (dual of [4782970, 4782857, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4782970 | 914−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- OOA 9-folding and stacking with additional row [i] based on linear OA(9113, 4782970, F9, 19) (dual of [4782970, 4782857, 20]-code), using
- net defined by OOA [i] based on linear OOA(9113, 531441, F9, 19, 19) (dual of [(531441, 19), 10097266, 20]-NRT-code), using
(207, 207+19, 7408912)-Net over F3 — Digital
Digital (207, 226, 7408912)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3226, 7408912, F3, 19) (dual of [7408912, 7408686, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3226, large, F3, 19) (dual of [large, large−226, 20]-code), using
- strength reduction [i] based on linear OA(3226, large, F3, 23) (dual of [large, large−226, 24]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,22], and designed minimum distance d ≥ |I|+1 = 24 [i]
- strength reduction [i] based on linear OA(3226, large, F3, 23) (dual of [large, large−226, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(3226, large, F3, 19) (dual of [large, large−226, 20]-code), using
(207, 207+19, large)-Net in Base 3 — Upper bound on s
There is no (207, 226, large)-net in base 3, because
- 17 times m-reduction [i] would yield (207, 209, large)-net in base 3, but