Best Known (158−24, 158, s)-Nets in Base 3
(158−24, 158, 1648)-Net over F3 — Constructive and digital
Digital (134, 158, 1648)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (2, 14, 8)-net over F3, using
- net from sequence [i] based on digital (2, 7)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 2 and N(F) ≥ 8, using
- net from sequence [i] based on digital (2, 7)-sequence over F3, using
- digital (120, 144, 1640)-net over F3, using
- net defined by OOA [i] based on linear OOA(3144, 1640, F3, 24, 24) (dual of [(1640, 24), 39216, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(3144, 19680, F3, 24) (dual of [19680, 19536, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(3144, 19683, F3, 24) (dual of [19683, 19539, 25]-code), using
- 1 times truncation [i] based on linear OA(3145, 19684, F3, 25) (dual of [19684, 19539, 26]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(3145, 19684, F3, 25) (dual of [19684, 19539, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3144, 19683, F3, 24) (dual of [19683, 19539, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(3144, 19680, F3, 24) (dual of [19680, 19536, 25]-code), using
- net defined by OOA [i] based on linear OOA(3144, 1640, F3, 24, 24) (dual of [(1640, 24), 39216, 25]-NRT-code), using
- digital (2, 14, 8)-net over F3, using
(158−24, 158, 11480)-Net over F3 — Digital
Digital (134, 158, 11480)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3158, 11480, F3, 24) (dual of [11480, 11322, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(3158, 19733, F3, 24) (dual of [19733, 19575, 25]-code), using
- 4 times code embedding in larger space [i] based on linear OA(3154, 19729, F3, 24) (dual of [19729, 19575, 25]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(3145, 19684, F3, 25) (dual of [19684, 19539, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(3109, 19684, F3, 19) (dual of [19684, 19575, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(39, 45, F3, 4) (dual of [45, 36, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- 4 times code embedding in larger space [i] based on linear OA(3154, 19729, F3, 24) (dual of [19729, 19575, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(3158, 19733, F3, 24) (dual of [19733, 19575, 25]-code), using
(158−24, 158, 5063221)-Net in Base 3 — Upper bound on s
There is no (134, 158, 5063222)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 2427 495632 397378 640665 794777 548905 792827 268306 093382 249528 338554 829754 542681 > 3158 [i]