Best Known (132, 132+37, s)-Nets in Base 3
(132, 132+37, 640)-Net over F3 — Constructive and digital
Digital (132, 169, 640)-net over F3, using
- 31 times duplication [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+37, 1325)-Net over F3 — Digital
Digital (132, 169, 1325)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3169, 1325, F3, 37) (dual of [1325, 1156, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(3169, 2187, F3, 37) (dual of [2187, 2018, 38]-code), using
- an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- discarding factors / shortening the dual code based on linear OA(3169, 2187, F3, 37) (dual of [2187, 2018, 38]-code), using
(132, 132+37, 107191)-Net in Base 3 — Upper bound on s
There is no (132, 169, 107192)-net in base 3, because
- 1 times m-reduction [i] would yield (132, 168, 107192)-net in base 3, but
- 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]