Best Known (227−45, 227, s)-Nets in Base 3
(227−45, 227, 688)-Net over F3 — Constructive and digital
Digital (182, 227, 688)-net over F3, using
- t-expansion [i] based on digital (181, 227, 688)-net over F3, using
- 5 times m-reduction [i] based on digital (181, 232, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 58, 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, 58, 172)-net over F81, using
- 5 times m-reduction [i] based on digital (181, 232, 688)-net over F3, using
(227−45, 227, 2522)-Net over F3 — Digital
Digital (182, 227, 2522)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3227, 2522, F3, 45) (dual of [2522, 2295, 46]-code), using
- 319 step Varšamov–Edel lengthening with (ri) = (3, 1, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 9 times 0, 1, 13 times 0, 1, 18 times 0, 1, 25 times 0, 1, 32 times 0, 1, 39 times 0, 1, 47 times 0, 1, 52 times 0, 1, 56 times 0) [i] based on linear OA(3210, 2186, F3, 45) (dual of [2186, 1976, 46]-code), using
- 1 times truncation [i] based on linear OA(3211, 2187, F3, 46) (dual of [2187, 1976, 47]-code), using
- an extension Ce(45) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,45], and designed minimum distance d ≥ |I|+1 = 46 [i]
- 1 times truncation [i] based on linear OA(3211, 2187, F3, 46) (dual of [2187, 1976, 47]-code), using
- 319 step Varšamov–Edel lengthening with (ri) = (3, 1, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 9 times 0, 1, 13 times 0, 1, 18 times 0, 1, 25 times 0, 1, 32 times 0, 1, 39 times 0, 1, 47 times 0, 1, 52 times 0, 1, 56 times 0) [i] based on linear OA(3210, 2186, F3, 45) (dual of [2186, 1976, 46]-code), using
(227−45, 227, 360689)-Net in Base 3 — Upper bound on s
There is no (182, 227, 360690)-net in base 3, because
- 1 times m-reduction [i] would yield (182, 226, 360690)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 675169 856020 208042 299048 573780 802356 341690 368559 516490 884637 207868 743873 662549 761288 281206 314423 513382 845309 > 3226 [i]