Best Known (116, 140, s)-Nets in Base 3
(116, 140, 700)-Net over F3 — Constructive and digital
Digital (116, 140, 700)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (4, 16, 12)-net over F3, using
- net from sequence [i] based on digital (4, 11)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 4 and N(F) ≥ 12, using
- net from sequence [i] based on digital (4, 11)-sequence over F3, using
- digital (100, 124, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 31, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 31, 172)-net over F81, using
- digital (4, 16, 12)-net over F3, using
(116, 140, 4660)-Net over F3 — Digital
Digital (116, 140, 4660)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3140, 4660, F3, 24) (dual of [4660, 4520, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(3140, 6605, F3, 24) (dual of [6605, 6465, 25]-code), using
- strength reduction [i] based on linear OA(3140, 6605, F3, 25) (dual of [6605, 6465, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(3129, 6562, F3, 25) (dual of [6562, 6433, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- 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(311, 43, F3, 5) (dual of [43, 32, 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([0,12]) ⊂ C([0,9]) [i] based on
- strength reduction [i] based on linear OA(3140, 6605, F3, 25) (dual of [6605, 6465, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3140, 6605, F3, 24) (dual of [6605, 6465, 25]-code), using
(116, 140, 974408)-Net in Base 3 — Upper bound on s
There is no (116, 140, 974409)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 6 265849 065603 279790 526397 114217 642889 493993 786099 612550 531790 004657 > 3140 [i]