Best Known (177−36, 177, s)-Nets in Base 3
(177−36, 177, 688)-Net over F3 — Constructive and digital
Digital (141, 177, 688)-net over F3, using
- 31 times duplication [i] based on digital (140, 176, 688)-net over F3, using
- t-expansion [i] based on digital (139, 176, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 44, 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, 44, 172)-net over F81, using
- t-expansion [i] based on digital (139, 176, 688)-net over F3, using
(177−36, 177, 1965)-Net over F3 — Digital
Digital (141, 177, 1965)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3177, 1965, F3, 36) (dual of [1965, 1788, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(3177, 2224, F3, 36) (dual of [2224, 2047, 37]-code), using
- construction X applied to C([0,18]) ⊂ C([0,15]) [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(3141, 2188, F3, 31) (dual of [2188, 2047, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (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([0,18]) ⊂ C([0,15]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3177, 2224, F3, 36) (dual of [2224, 2047, 37]-code), using
(177−36, 177, 185674)-Net in Base 3 — Upper bound on s
There is no (141, 177, 185675)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 2 821612 759693 769910 743097 093248 650510 049286 679033 932276 792700 429060 741710 391440 712981 > 3177 [i]