Best Known (219−38, 219, s)-Nets in Base 3
(219−38, 219, 1480)-Net over F3 — Constructive and digital
Digital (181, 219, 1480)-net over F3, using
- 1 times m-reduction [i] based on digital (181, 220, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 55, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- trace code for nets [i] based on digital (16, 55, 370)-net over F81, using
(219−38, 219, 5498)-Net over F3 — Digital
Digital (181, 219, 5498)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3219, 5498, F3, 38) (dual of [5498, 5279, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(3219, 6619, F3, 38) (dual of [6619, 6400, 39]-code), using
- 3 times code embedding in larger space [i] based on linear OA(3216, 6616, F3, 38) (dual of [6616, 6400, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(30) [i] based on
- linear OA(3201, 6561, F3, 38) (dual of [6561, 6360, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- 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(315, 55, F3, 6) (dual of [55, 40, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(315, 85, F3, 6) (dual of [85, 70, 7]-code), using
- construction X applied to Ce(37) ⊂ Ce(30) [i] based on
- 3 times code embedding in larger space [i] based on linear OA(3216, 6616, F3, 38) (dual of [6616, 6400, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(3219, 6619, F3, 38) (dual of [6619, 6400, 39]-code), using
(219−38, 219, 1252072)-Net in Base 3 — Upper bound on s
There is no (181, 219, 1252073)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 308 717020 501854 277762 294697 098962 395079 622544 939838 545095 949294 233574 548020 492901 230728 029823 276325 201339 > 3219 [i]