Best Known (156−18, 156, s)-Nets in Base 3
(156−18, 156, 177147)-Net over F3 — Constructive and digital
Digital (138, 156, 177147)-net over F3, using
- net defined by OOA [i] based on linear OOA(3156, 177147, F3, 18, 18) (dual of [(177147, 18), 3188490, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(3156, 1594323, F3, 18) (dual of [1594323, 1594167, 19]-code), using
- 1 times truncation [i] based on linear OA(3157, 1594324, F3, 19) (dual of [1594324, 1594167, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 1594324 | 326−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(3157, 1594324, F3, 19) (dual of [1594324, 1594167, 20]-code), using
- OA 9-folding and stacking [i] based on linear OA(3156, 1594323, F3, 18) (dual of [1594323, 1594167, 19]-code), using
(156−18, 156, 495074)-Net over F3 — Digital
Digital (138, 156, 495074)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3156, 495074, F3, 3, 18) (dual of [(495074, 3), 1485066, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3156, 531441, F3, 3, 18) (dual of [(531441, 3), 1594167, 19]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3156, 1594323, F3, 18) (dual of [1594323, 1594167, 19]-code), using
- 1 times truncation [i] based on linear OA(3157, 1594324, F3, 19) (dual of [1594324, 1594167, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 1594324 | 326−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(3157, 1594324, F3, 19) (dual of [1594324, 1594167, 20]-code), using
- OOA 3-folding [i] based on linear OA(3156, 1594323, F3, 18) (dual of [1594323, 1594167, 19]-code), using
- discarding factors / shortening the dual code based on linear OOA(3156, 531441, F3, 3, 18) (dual of [(531441, 3), 1594167, 19]-NRT-code), using
(156−18, 156, large)-Net in Base 3 — Upper bound on s
There is no (138, 156, large)-net in base 3, because
- 16 times m-reduction [i] would yield (138, 140, large)-net in base 3, but