Best Known (219−43, 219, s)-Nets in Base 3
(219−43, 219, 688)-Net over F3 — Constructive and digital
Digital (176, 219, 688)-net over F3, using
- t-expansion [i] based on digital (175, 219, 688)-net over F3, using
- 5 times m-reduction [i] based on digital (175, 224, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 56, 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, 56, 172)-net over F81, using
- 5 times m-reduction [i] based on digital (175, 224, 688)-net over F3, using
(219−43, 219, 2567)-Net over F3 — Digital
Digital (176, 219, 2567)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3219, 2567, F3, 43) (dual of [2567, 2348, 44]-code), using
- 357 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 7 times 0, 1, 10 times 0, 1, 13 times 0, 1, 17 times 0, 1, 22 times 0, 1, 27 times 0, 1, 33 times 0, 1, 40 times 0, 1, 47 times 0, 1, 53 times 0, 1, 58 times 0) [i] based on linear OA(3197, 2188, F3, 43) (dual of [2188, 1991, 44]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,21], and minimum distance d ≥ |{−21,−20,…,21}|+1 = 44 (BCH-bound) [i]
- 357 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 7 times 0, 1, 10 times 0, 1, 13 times 0, 1, 17 times 0, 1, 22 times 0, 1, 27 times 0, 1, 33 times 0, 1, 40 times 0, 1, 47 times 0, 1, 53 times 0, 1, 58 times 0) [i] based on linear OA(3197, 2188, F3, 43) (dual of [2188, 1991, 44]-code), using
(219−43, 219, 389415)-Net in Base 3 — Upper bound on s
There is no (176, 219, 389416)-net in base 3, because
- 1 times m-reduction [i] would yield (176, 218, 389416)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 102 909556 944079 789953 003649 402705 644175 332420 325617 489012 358741 396316 785790 495007 112826 013668 754774 244049 > 3218 [i]