Best Known (163, 163+42, s)-Nets in Base 3
(163, 163+42, 688)-Net over F3 — Constructive and digital
Digital (163, 205, 688)-net over F3, using
- 3 times m-reduction [i] based on digital (163, 208, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 52, 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, 52, 172)-net over F81, using
(163, 163+42, 2101)-Net over F3 — Digital
Digital (163, 205, 2101)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3205, 2101, F3, 42) (dual of [2101, 1896, 43]-code), using
- discarding factors / shortening the dual code based on linear OA(3205, 2224, F3, 42) (dual of [2224, 2019, 43]-code), using
- construction X applied to C([0,21]) ⊂ C([0,18]) [i] based on
- linear OA(3197, 2188, F3, 43) (dual of [2188, 1991, 44]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,21], and minimum distance d ≥ |{−21,−20,…,21}|+1 = 44 (BCH-bound) [i]
- linear OA(3169, 2188, F3, 37) (dual of [2188, 2019, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(38, 36, F3, 4) (dual of [36, 28, 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([0,21]) ⊂ C([0,18]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3205, 2224, F3, 42) (dual of [2224, 2019, 43]-code), using
(163, 163+42, 197255)-Net in Base 3 — Upper bound on s
There is no (163, 205, 197256)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 64 545860 530525 121531 195558 953250 657414 443783 355579 367523 036679 185810 311980 734177 746006 587937 111441 > 3205 [i]