Best Known (141−27, 141, s)-Nets in Base 3
(141−27, 141, 688)-Net over F3 — Constructive and digital
Digital (114, 141, 688)-net over F3, using
- 31 times duplication [i] based on digital (113, 140, 688)-net over F3, using
- t-expansion [i] based on digital (112, 140, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 35, 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, 35, 172)-net over F81, using
- t-expansion [i] based on digital (112, 140, 688)-net over F3, using
(141−27, 141, 2306)-Net over F3 — Digital
Digital (114, 141, 2306)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3141, 2306, F3, 27) (dual of [2306, 2165, 28]-code), using
- 105 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 8 times 0, 1, 11 times 0, 1, 15 times 0, 1, 20 times 0, 1, 26 times 0) [i] based on linear OA(3126, 2186, F3, 27) (dual of [2186, 2060, 28]-code), using
- 1 times truncation [i] based on linear OA(3127, 2187, F3, 28) (dual of [2187, 2060, 29]-code), using
- an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- 1 times truncation [i] based on linear OA(3127, 2187, F3, 28) (dual of [2187, 2060, 29]-code), using
- 105 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 8 times 0, 1, 11 times 0, 1, 15 times 0, 1, 20 times 0, 1, 26 times 0) [i] based on linear OA(3126, 2186, F3, 27) (dual of [2186, 2060, 28]-code), using
(141−27, 141, 389575)-Net in Base 3 — Upper bound on s
There is no (114, 141, 389576)-net in base 3, because
- 1 times m-reduction [i] would yield (114, 140, 389576)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 6 265984 244806 503254 644311 502786 107442 104126 738991 132168 902809 040273 > 3140 [i]