Best Known (144, 181, s)-Nets in Base 3
(144, 181, 688)-Net over F3 — Constructive and digital
Digital (144, 181, 688)-net over F3, using
- 31 times duplication [i] based on digital (143, 180, 688)-net over F3, using
- t-expansion [i] based on digital (142, 180, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 45, 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, 45, 172)-net over F81, using
- t-expansion [i] based on digital (142, 180, 688)-net over F3, using
(144, 181, 1944)-Net over F3 — Digital
Digital (144, 181, 1944)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3181, 1944, F3, 37) (dual of [1944, 1763, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(3181, 2228, F3, 37) (dual of [2228, 2047, 38]-code), using
- 2 times code embedding in larger space [i] based on linear OA(3179, 2226, F3, 37) (dual of [2226, 2047, 38]-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(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([0,18]) ⊂ C([0,15]) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(3179, 2226, F3, 37) (dual of [2226, 2047, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(3181, 2228, F3, 37) (dual of [2228, 2047, 38]-code), using
(144, 181, 222986)-Net in Base 3 — Upper bound on s
There is no (144, 181, 222987)-net in base 3, because
- 1 times m-reduction [i] would yield (144, 180, 222987)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 76 180560 388871 196698 096650 667316 510287 747857 431120 500365 398234 688305 675671 265886 764181 > 3180 [i]