Best Known (96−13, 96, s)-Nets in Base 3
(96−13, 96, 29531)-Net over F3 — Constructive and digital
Digital (83, 96, 29531)-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 (76, 89, 29524)-net over F3, using
- net defined by OOA [i] based on linear OOA(389, 29524, F3, 13, 13) (dual of [(29524, 13), 383723, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(389, 177145, F3, 13) (dual of [177145, 177056, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(389, 177147, F3, 13) (dual of [177147, 177058, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(389, 177147, F3, 13) (dual of [177147, 177058, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(389, 177145, F3, 13) (dual of [177145, 177056, 14]-code), using
- net defined by OOA [i] based on linear OOA(389, 29524, F3, 13, 13) (dual of [(29524, 13), 383723, 14]-NRT-code), using
- digital (1, 7, 7)-net over F3, using
(96−13, 96, 69156)-Net over F3 — Digital
Digital (83, 96, 69156)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(396, 69156, F3, 2, 13) (dual of [(69156, 2), 138216, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(396, 88588, F3, 2, 13) (dual of [(88588, 2), 177080, 14]-NRT-code), using
- 31 times duplication [i] based on linear OOA(395, 88588, F3, 2, 13) (dual of [(88588, 2), 177081, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(395, 177176, F3, 13) (dual of [177176, 177081, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- linear OA(389, 177148, F3, 13) (dual of [177148, 177059, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(367, 177148, F3, 9) (dual of [177148, 177081, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(36, 28, F3, 3) (dual of [28, 22, 4]-code or 28-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 2-folding [i] based on linear OA(395, 177176, F3, 13) (dual of [177176, 177081, 14]-code), using
- 31 times duplication [i] based on linear OOA(395, 88588, F3, 2, 13) (dual of [(88588, 2), 177081, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(396, 88588, F3, 2, 13) (dual of [(88588, 2), 177080, 14]-NRT-code), using
(96−13, 96, large)-Net in Base 3 — Upper bound on s
There is no (83, 96, large)-net in base 3, because
- 11 times m-reduction [i] would yield (83, 85, large)-net in base 3, but