Best Known (164, 164+42, s)-Nets in Base 3
(164, 164+42, 688)-Net over F3 — Constructive and digital
Digital (164, 206, 688)-net over F3, using
- t-expansion [i] based on digital (163, 206, 688)-net over F3, using
- 2 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
- 2 times m-reduction [i] based on digital (163, 208, 688)-net over F3, using
(164, 164+42, 2161)-Net over F3 — Digital
Digital (164, 206, 2161)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3206, 2161, F3, 42) (dual of [2161, 1955, 43]-code), using
- discarding factors / shortening the dual code based on linear OA(3206, 2225, F3, 42) (dual of [2225, 2019, 43]-code), using
- 1 times truncation [i] based on linear OA(3207, 2226, F3, 43) (dual of [2226, 2019, 44]-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(310, 38, F3, 5) (dual of [38, 28, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(310, 39, F3, 5) (dual of [39, 29, 6]-code), using
- construction X applied to C([0,21]) ⊂ C([0,18]) [i] based on
- 1 times truncation [i] based on linear OA(3207, 2226, F3, 43) (dual of [2226, 2019, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(3206, 2225, F3, 42) (dual of [2225, 2019, 43]-code), using
(164, 164+42, 207850)-Net in Base 3 — Upper bound on s
There is no (164, 206, 207851)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 193 633238 380613 787954 086886 888604 962355 535919 844351 244766 244804 957779 184476 381455 682252 580527 140463 > 3206 [i]