Best Known (148, 148+18, s)-Nets in Base 3
(148, 148+18, 177154)-Net over F3 — Constructive and digital
Digital (148, 166, 177154)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 10, 7)-net over F3, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 1 and N(F) ≥ 7, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- 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
- net defined by OOA [i] based on linear OOA(3156, 177147, F3, 18, 18) (dual of [(177147, 18), 3188490, 19]-NRT-code), using
- digital (1, 10, 7)-net over F3, using
(148, 148+18, 531461)-Net over F3 — Digital
Digital (148, 166, 531461)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3166, 531461, F3, 3, 18) (dual of [(531461, 3), 1594217, 19]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3166, 1594383, F3, 18) (dual of [1594383, 1594217, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(3166, 1594385, F3, 18) (dual of [1594385, 1594219, 19]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [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]
- linear OA(3105, 1594324, F3, 13) (dual of [1594324, 1594219, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 1594324 | 326−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(39, 61, F3, 4) (dual of [61, 52, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3166, 1594385, F3, 18) (dual of [1594385, 1594219, 19]-code), using
- OOA 3-folding [i] based on linear OA(3166, 1594383, F3, 18) (dual of [1594383, 1594217, 19]-code), using
(148, 148+18, large)-Net in Base 3 — Upper bound on s
There is no (148, 166, large)-net in base 3, because
- 16 times m-reduction [i] would yield (148, 150, large)-net in base 3, but