Best Known (132, 132+36, s)-Nets in Base 3
(132, 132+36, 640)-Net over F3 — Constructive and digital
Digital (132, 168, 640)-net over F3, using
- t-expansion [i] based on digital (131, 168, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 42, 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, 42, 160)-net over F81, using
(132, 132+36, 1461)-Net over F3 — Digital
Digital (132, 168, 1461)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3168, 1461, F3, 36) (dual of [1461, 1293, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(3168, 2187, F3, 36) (dual of [2187, 2019, 37]-code), using
- 1 times truncation [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]
- 1 times truncation [i] based on linear OA(3169, 2188, F3, 37) (dual of [2188, 2019, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(3168, 2187, F3, 36) (dual of [2187, 2019, 37]-code), using
(132, 132+36, 107191)-Net in Base 3 — Upper bound on s
There is no (132, 168, 107192)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 143 349995 862129 357124 952276 099132 016704 866756 189691 807942 588297 690338 722094 963665 > 3168 [i]