Best Known (88, 106, s)-Nets in Base 3
(88, 106, 736)-Net over F3 — Constructive and digital
Digital (88, 106, 736)-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 (78, 96, 729)-net over F3, using
- net defined by OOA [i] based on linear OOA(396, 729, F3, 18, 18) (dual of [(729, 18), 13026, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(396, 6561, F3, 18) (dual of [6561, 6465, 19]-code), using
- 1 times truncation [i] based on linear OA(397, 6562, F3, 19) (dual of [6562, 6465, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, 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(397, 6562, F3, 19) (dual of [6562, 6465, 20]-code), using
- OA 9-folding and stacking [i] based on linear OA(396, 6561, F3, 18) (dual of [6561, 6465, 19]-code), using
- net defined by OOA [i] based on linear OOA(396, 729, F3, 18, 18) (dual of [(729, 18), 13026, 19]-NRT-code), using
- digital (1, 10, 7)-net over F3, using
(88, 106, 4584)-Net over F3 — Digital
Digital (88, 106, 4584)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3106, 4584, F3, 18) (dual of [4584, 4478, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(3106, 6603, F3, 18) (dual of [6603, 6497, 19]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3105, 6602, F3, 18) (dual of [6602, 6497, 19]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- linear OA(397, 6562, F3, 19) (dual of [6562, 6465, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(365, 6562, F3, 13) (dual of [6562, 6497, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(38, 40, F3, 4) (dual of [40, 32, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3105, 6602, F3, 18) (dual of [6602, 6497, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(3106, 6603, F3, 18) (dual of [6603, 6497, 19]-code), using
(88, 106, 863267)-Net in Base 3 — Upper bound on s
There is no (88, 106, 863268)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 375 712977 962204 111280 704700 740822 187959 939545 943881 > 3106 [i]