Best Known (193, 193+45, s)-Nets in Base 3
(193, 193+45, 896)-Net over F3 — Constructive and digital
Digital (193, 238, 896)-net over F3, using
- 2 times m-reduction [i] based on digital (193, 240, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 60, 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, 60, 224)-net over F81, using
(193, 193+45, 3342)-Net over F3 — Digital
Digital (193, 238, 3342)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3238, 3342, F3, 45) (dual of [3342, 3104, 46]-code), using
- 60 step Varšamov–Edel lengthening with (ri) = (1, 19 times 0, 1, 39 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
- 60 step Varšamov–Edel lengthening with (ri) = (1, 19 times 0, 1, 39 times 0) [i] based on linear OA(3236, 3280, F3, 45) (dual of [3280, 3044, 46]-code), using
(193, 193+45, 624748)-Net in Base 3 — Upper bound on s
There is no (193, 238, 624749)-net in base 3, because
- 1 times m-reduction [i] would yield (193, 237, 624749)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 119603 706937 420405 090054 142796 421544 255441 133145 459844 301284 671507 493229 746026 790845 394581 237388 985125 101467 520657 > 3237 [i]