Best Known (157−19, 157, s)-Nets in Base 3
(157−19, 157, 177147)-Net over F3 — Constructive and digital
Digital (138, 157, 177147)-net over F3, using
- net defined by OOA [i] based on linear OOA(3157, 177147, F3, 19, 19) (dual of [(177147, 19), 3365636, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [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]
- OOA 9-folding and stacking with additional row [i] based on linear OA(3157, 1594324, F3, 19) (dual of [1594324, 1594167, 20]-code), using
(157−19, 157, 398581)-Net over F3 — Digital
Digital (138, 157, 398581)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3157, 398581, F3, 4, 19) (dual of [(398581, 4), 1594167, 20]-NRT-code), using
- OOA 4-folding [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]
- OOA 4-folding [i] based on linear OA(3157, 1594324, F3, 19) (dual of [1594324, 1594167, 20]-code), using
(157−19, 157, large)-Net in Base 3 — Upper bound on s
There is no (138, 157, large)-net in base 3, because
- 17 times m-reduction [i] would yield (138, 140, large)-net in base 3, but