Best Known (189−38, 189, s)-Nets in Base 3
(189−38, 189, 688)-Net over F3 — Constructive and digital
Digital (151, 189, 688)-net over F3, using
- 3 times m-reduction [i] based on digital (151, 192, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 48, 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, 48, 172)-net over F81, using
(189−38, 189, 2181)-Net over F3 — Digital
Digital (151, 189, 2181)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3189, 2181, F3, 38) (dual of [2181, 1992, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(3189, 2229, F3, 38) (dual of [2229, 2040, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(30) [i] based on
- linear OA(3176, 2187, F3, 38) (dual of [2187, 2011, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(3141, 2187, F3, 31) (dual of [2187, 2046, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(313, 42, F3, 6) (dual of [42, 29, 7]-code), using
- construction X applied to Ce(37) ⊂ Ce(30) [i] based on
- discarding factors / shortening the dual code based on linear OA(3189, 2229, F3, 38) (dual of [2229, 2040, 39]-code), using
(189−38, 189, 220927)-Net in Base 3 — Upper bound on s
There is no (151, 189, 220928)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 1 499445 455452 097145 849606 094335 637933 711701 611267 518420 049959 908987 768613 241151 312149 322753 > 3189 [i]