Best Known (187, 187+48, s)-Nets in Base 3
(187, 187+48, 688)-Net over F3 — Constructive and digital
Digital (187, 235, 688)-net over F3, using
- 5 times m-reduction [i] based on digital (187, 240, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 60, 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, 60, 172)-net over F81, using
(187, 187+48, 2295)-Net over F3 — Digital
Digital (187, 235, 2295)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3235, 2295, F3, 48) (dual of [2295, 2060, 49]-code), using
- 97 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 10 times 0, 1, 15 times 0, 1, 21 times 0, 1, 28 times 0) [i] based on linear OA(3224, 2187, F3, 48) (dual of [2187, 1963, 49]-code), using
- 1 times truncation [i] based on linear OA(3225, 2188, F3, 49) (dual of [2188, 1963, 50]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,24], and minimum distance d ≥ |{−24,−23,…,24}|+1 = 50 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(3225, 2188, F3, 49) (dual of [2188, 1963, 50]-code), using
- 97 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 10 times 0, 1, 15 times 0, 1, 21 times 0, 1, 28 times 0) [i] based on linear OA(3224, 2187, F3, 48) (dual of [2187, 1963, 49]-code), using
(187, 187+48, 230197)-Net in Base 3 — Upper bound on s
There is no (187, 235, 230198)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 13289 703192 436910 214701 780386 784714 492771 130280 939675 192372 813886 087921 425915 307032 125253 495533 019163 184132 127089 > 3235 [i]