Best Known (175−37, 175, s)-Nets in Base 3
(175−37, 175, 640)-Net over F3 — Constructive and digital
Digital (138, 175, 640)-net over F3, using
- t-expansion [i] based on digital (137, 175, 640)-net over F3, using
- 1 times m-reduction [i] based on digital (137, 176, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 44, 160)-net over F81, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 5 and N(F) ≥ 160, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- trace code for nets [i] based on digital (5, 44, 160)-net over F81, using
- 1 times m-reduction [i] based on digital (137, 176, 640)-net over F3, using
(175−37, 175, 1605)-Net over F3 — Digital
Digital (138, 175, 1605)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3175, 1605, F3, 37) (dual of [1605, 1430, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(3175, 2208, F3, 37) (dual of [2208, 2033, 38]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3174, 2207, F3, 37) (dual of [2207, 2033, 38]-code), using
- construction X applied to C([0,18]) ⊂ C([0,16]) [i] based on
- 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(3155, 2188, F3, 33) (dual of [2188, 2033, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(35, 19, F3, 3) (dual of [19, 14, 4]-code or 19-cap in PG(4,3)), using
- construction X applied to C([0,18]) ⊂ C([0,16]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3174, 2207, F3, 37) (dual of [2207, 2033, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(3175, 2208, F3, 37) (dual of [2208, 2033, 38]-code), using
(175−37, 175, 154604)-Net in Base 3 — Upper bound on s
There is no (138, 175, 154605)-net in base 3, because
- 1 times m-reduction [i] would yield (138, 174, 154605)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 104497 835522 235830 987910 671103 296792 212476 007606 578198 812427 457506 250303 501858 437537 > 3174 [i]