Best Known (115, 115+24, s)-Nets in Base 3
(115, 115+24, 698)-Net over F3 — Constructive and digital
Digital (115, 139, 698)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (3, 15, 10)-net over F3, using
- net from sequence [i] based on digital (3, 9)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 3 and N(F) ≥ 10, using
- net from sequence [i] based on digital (3, 9)-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 (3, 15, 10)-net over F3, using
(115, 115+24, 4432)-Net over F3 — Digital
Digital (115, 139, 4432)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3139, 4432, F3, 24) (dual of [4432, 4293, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(3139, 6604, F3, 24) (dual of [6604, 6465, 25]-code), using
- 1 times truncation [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
- 1 times truncation [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(3139, 6604, F3, 24) (dual of [6604, 6465, 25]-code), using
(115, 115+24, 889160)-Net in Base 3 — Upper bound on s
There is no (115, 139, 889161)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 2 088599 905103 849849 682904 087889 142040 189970 868952 731891 638970 235313 > 3139 [i]