Best Known (250−38, 250, s)-Nets in Base 3
(250−38, 250, 1508)-Net over F3 — Constructive and digital
Digital (212, 250, 1508)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (15, 34, 28)-net over F3, using
- net from sequence [i] based on digital (15, 27)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 15 and N(F) ≥ 28, using
- net from sequence [i] based on digital (15, 27)-sequence over F3, using
- digital (178, 216, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 54, 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, 54, 370)-net over F81, using
- digital (15, 34, 28)-net over F3, using
(250−38, 250, 14215)-Net over F3 — Digital
Digital (212, 250, 14215)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3250, 14215, F3, 38) (dual of [14215, 13965, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(3250, 19763, F3, 38) (dual of [19763, 19513, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(27) [i] based on
- linear OA(3226, 19683, F3, 38) (dual of [19683, 19457, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(3163, 19683, F3, 28) (dual of [19683, 19520, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(324, 80, F3, 9) (dual of [80, 56, 10]-code), using
- the primitive narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- construction X applied to Ce(37) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(3250, 19763, F3, 38) (dual of [19763, 19513, 39]-code), using
(250−38, 250, 7517885)-Net in Base 3 — Upper bound on s
There is no (212, 250, 7517886)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 190684 144694 851176 841711 351857 761002 764228 441049 047148 536065 660028 039791 975491 447438 517165 862394 561532 246294 895865 359201 > 3250 [i]