Best Known (175−30, 175, s)-Nets in Base 3
(175−30, 175, 896)-Net over F3 — Constructive and digital
Digital (145, 175, 896)-net over F3, using
- 1 times m-reduction [i] based on digital (145, 176, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 44, 224)-net over F81, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 13 and N(F) ≥ 224, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- trace code for nets [i] based on digital (13, 44, 224)-net over F81, using
(175−30, 175, 5185)-Net over F3 — Digital
Digital (145, 175, 5185)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3175, 5185, F3, 30) (dual of [5185, 5010, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(3175, 6614, F3, 30) (dual of [6614, 6439, 31]-code), using
- construction X applied to Ce(30) ⊂ Ce(22) [i] based on
- linear OA(3161, 6561, F3, 31) (dual of [6561, 6400, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3121, 6561, F3, 23) (dual of [6561, 6440, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(314, 53, F3, 6) (dual of [53, 39, 7]-code), using
- construction X applied to Ce(30) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(3175, 6614, F3, 30) (dual of [6614, 6439, 31]-code), using
(175−30, 175, 1183439)-Net in Base 3 — Upper bound on s
There is no (145, 175, 1183440)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 313490 080904 175543 802582 286713 775301 179726 451772 699921 170983 429976 250278 901784 507841 > 3175 [i]