Best Known (160, 160+39, s)-Nets in Base 3
(160, 160+39, 688)-Net over F3 — Constructive and digital
Digital (160, 199, 688)-net over F3, using
- 5 times m-reduction [i] based on digital (160, 204, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 51, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 51, 172)-net over F81, using
(160, 160+39, 2421)-Net over F3 — Digital
Digital (160, 199, 2421)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3199, 2421, F3, 39) (dual of [2421, 2222, 40]-code), using
- 218 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 10 times 0, 1, 15 times 0, 1, 19 times 0, 1, 25 times 0, 1, 32 times 0, 1, 40 times 0, 1, 47 times 0) [i] based on linear OA(3182, 2186, F3, 39) (dual of [2186, 2004, 40]-code), using
- 1 times truncation [i] based on linear OA(3183, 2187, F3, 40) (dual of [2187, 2004, 41]-code), using
- an extension Ce(39) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- 1 times truncation [i] based on linear OA(3183, 2187, F3, 40) (dual of [2187, 2004, 41]-code), using
- 218 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 10 times 0, 1, 15 times 0, 1, 19 times 0, 1, 25 times 0, 1, 32 times 0, 1, 40 times 0, 1, 47 times 0) [i] based on linear OA(3182, 2186, F3, 39) (dual of [2186, 2004, 40]-code), using
(160, 160+39, 371765)-Net in Base 3 — Upper bound on s
There is no (160, 199, 371766)-net in base 3, because
- 1 times m-reduction [i] would yield (160, 198, 371766)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 29512 744806 530762 616328 144990 665705 087873 812702 653862 426650 145346 366210 823061 318399 596180 302465 > 3198 [i]