Best Known (112−13, 112, s)-Nets in Base 3
(112−13, 112, 265727)-Net over F3 — Constructive and digital
Digital (99, 112, 265727)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 7, 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
- a shift-net [i]
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- digital (92, 105, 265720)-net over F3, using
- net defined by OOA [i] based on linear OOA(3105, 265720, F3, 13, 13) (dual of [(265720, 13), 3454255, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3105, 1594321, F3, 13) (dual of [1594321, 1594216, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(3105, 1594323, F3, 13) (dual of [1594323, 1594218, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(3105, 1594323, F3, 13) (dual of [1594323, 1594218, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3105, 1594321, F3, 13) (dual of [1594321, 1594216, 14]-code), using
- net defined by OOA [i] based on linear OOA(3105, 265720, F3, 13, 13) (dual of [(265720, 13), 3454255, 14]-NRT-code), using
- digital (1, 7, 7)-net over F3, using
(112−13, 112, 531452)-Net over F3 — Digital
Digital (99, 112, 531452)-net over F3, using
- 31 times duplication [i] based on digital (98, 111, 531452)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3111, 531452, F3, 3, 13) (dual of [(531452, 3), 1594245, 14]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3111, 1594356, F3, 13) (dual of [1594356, 1594245, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- 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(379, 1594324, F3, 9) (dual of [1594324, 1594245, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 1594324 | 326−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(36, 32, F3, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- OOA 3-folding [i] based on linear OA(3111, 1594356, F3, 13) (dual of [1594356, 1594245, 14]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3111, 531452, F3, 3, 13) (dual of [(531452, 3), 1594245, 14]-NRT-code), using
(112−13, 112, large)-Net in Base 3 — Upper bound on s
There is no (99, 112, large)-net in base 3, because
- 11 times m-reduction [i] would yield (99, 101, large)-net in base 3, but