Best Known (167−18, 167, s)-Nets in Base 3
(167−18, 167, 177155)-Net over F3 — Constructive and digital
Digital (149, 167, 177155)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (2, 11, 8)-net over F3, using
- net from sequence [i] based on digital (2, 7)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 2 and N(F) ≥ 8, using
- net from sequence [i] based on digital (2, 7)-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 (2, 11, 8)-net over F3, using
(167−18, 167, 568931)-Net over F3 — Digital
Digital (149, 167, 568931)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3167, 568931, F3, 2, 18) (dual of [(568931, 2), 1137695, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3167, 797193, F3, 2, 18) (dual of [(797193, 2), 1594219, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3167, 1594386, F3, 18) (dual of [1594386, 1594219, 19]-code), using
- 1 times truncation [i] based on linear OA(3168, 1594387, F3, 19) (dual of [1594387, 1594219, 20]-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(311, 63, F3, 5) (dual of [63, 52, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(311, 85, F3, 5) (dual of [85, 74, 6]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- 1 times truncation [i] based on linear OA(3168, 1594387, F3, 19) (dual of [1594387, 1594219, 20]-code), using
- OOA 2-folding [i] based on linear OA(3167, 1594386, F3, 18) (dual of [1594386, 1594219, 19]-code), using
- discarding factors / shortening the dual code based on linear OOA(3167, 797193, F3, 2, 18) (dual of [(797193, 2), 1594219, 19]-NRT-code), using
(167−18, 167, large)-Net in Base 3 — Upper bound on s
There is no (149, 167, large)-net in base 3, because
- 16 times m-reduction [i] would yield (149, 151, large)-net in base 3, but