Best Known (167−19, 167, s)-Nets in Base 3
(167−19, 167, 177154)-Net over F3 — Constructive and digital
Digital (148, 167, 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, 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
- net defined by OOA [i] based on linear OOA(3157, 177147, F3, 19, 19) (dual of [(177147, 19), 3365636, 20]-NRT-code), using
- digital (1, 10, 7)-net over F3, using
(167−19, 167, 528751)-Net over F3 — Digital
Digital (148, 167, 528751)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3167, 528751, F3, 3, 19) (dual of [(528751, 3), 1586086, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3167, 531457, F3, 3, 19) (dual of [(531457, 3), 1594204, 20]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3166, 531457, F3, 3, 19) (dual of [(531457, 3), 1594205, 20]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3166, 1594371, F3, 19) (dual of [1594371, 1594205, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- linear OA(3157, 1594323, F3, 19) (dual of [1594323, 1594166, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(3118, 1594323, F3, 14) (dual of [1594323, 1594205, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(39, 48, F3, 4) (dual of [48, 39, 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 Ce(18) ⊂ Ce(13) [i] based on
- OOA 3-folding [i] based on linear OA(3166, 1594371, F3, 19) (dual of [1594371, 1594205, 20]-code), using
- 31 times duplication [i] based on linear OOA(3166, 531457, F3, 3, 19) (dual of [(531457, 3), 1594205, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3167, 531457, F3, 3, 19) (dual of [(531457, 3), 1594204, 20]-NRT-code), using
(167−19, 167, large)-Net in Base 3 — Upper bound on s
There is no (148, 167, large)-net in base 3, because
- 17 times m-reduction [i] would yield (148, 150, large)-net in base 3, but