Best Known (186−33, 186, s)-Nets in Base 3
(186−33, 186, 896)-Net over F3 — Constructive and digital
Digital (153, 186, 896)-net over F3, using
- 32 times duplication [i] based on digital (151, 184, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 46, 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, 46, 224)-net over F81, using
(186−33, 186, 4339)-Net over F3 — Digital
Digital (153, 186, 4339)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3186, 4339, F3, 33) (dual of [4339, 4153, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3186, 6602, F3, 33) (dual of [6602, 6416, 34]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3185, 6601, F3, 33) (dual of [6601, 6416, 34]-code), using
- construction X applied to Ce(33) ⊂ Ce(27) [i] based on
- linear OA(3177, 6561, F3, 34) (dual of [6561, 6384, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(3145, 6561, F3, 28) (dual of [6561, 6416, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(38, 40, F3, 4) (dual of [40, 32, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- construction X applied to Ce(33) ⊂ Ce(27) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3185, 6601, F3, 33) (dual of [6601, 6416, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3186, 6602, F3, 33) (dual of [6602, 6416, 34]-code), using
(186−33, 186, 1117421)-Net in Base 3 — Upper bound on s
There is no (153, 186, 1117422)-net in base 3, because
- 1 times m-reduction [i] would yield (153, 185, 1117422)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 18511 099016 277219 517935 806326 247270 592032 178921 708694 278497 877966 447825 550447 137457 474465 > 3185 [i]