Best Known (192, 192+45, s)-Nets in Base 3
(192, 192+45, 896)-Net over F3 — Constructive and digital
Digital (192, 237, 896)-net over F3, using
- 31 times duplication [i] based on digital (191, 236, 896)-net over F3, using
- t-expansion [i] based on digital (190, 236, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 59, 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, 59, 224)-net over F81, using
- t-expansion [i] based on digital (190, 236, 896)-net over F3, using
(192, 192+45, 3301)-Net over F3 — Digital
Digital (192, 237, 3301)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3237, 3301, F3, 45) (dual of [3301, 3064, 46]-code), using
- 20 step Varšamov–Edel lengthening with (ri) = (1, 19 times 0) [i] based on linear OA(3236, 3280, F3, 45) (dual of [3280, 3044, 46]-code), using
- 1 times truncation [i] based on linear OA(3237, 3281, F3, 46) (dual of [3281, 3044, 47]-code), using
- an extension Ce(45) of the narrow-sense BCH-code C(I) with length 3280 | 38−1, defining interval I = [1,45], and designed minimum distance d ≥ |I|+1 = 46 [i]
- 1 times truncation [i] based on linear OA(3237, 3281, F3, 46) (dual of [3281, 3044, 47]-code), using
- 20 step Varšamov–Edel lengthening with (ri) = (1, 19 times 0) [i] based on linear OA(3236, 3280, F3, 45) (dual of [3280, 3044, 46]-code), using
(192, 192+45, 594315)-Net in Base 3 — Upper bound on s
There is no (192, 237, 594316)-net in base 3, because
- 1 times m-reduction [i] would yield (192, 236, 594316)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 39867 796784 922407 253848 552682 750146 458985 134976 195260 276177 488188 139615 671481 136844 883949 677980 606260 766151 283977 > 3236 [i]