Best Known (194, 194+49, s)-Nets in Base 3
(194, 194+49, 688)-Net over F3 — Constructive and digital
Digital (194, 243, 688)-net over F3, using
- t-expansion [i] based on digital (193, 243, 688)-net over F3, using
- 5 times m-reduction [i] based on digital (193, 248, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 62, 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, 62, 172)-net over F81, using
- 5 times m-reduction [i] based on digital (193, 248, 688)-net over F3, using
(194, 194+49, 2469)-Net over F3 — Digital
Digital (194, 243, 2469)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3243, 2469, F3, 49) (dual of [2469, 2226, 50]-code), using
- 263 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 7 times 0, 1, 11 times 0, 1, 15 times 0, 1, 20 times 0, 1, 25 times 0, 1, 32 times 0, 1, 38 times 0, 1, 43 times 0, 1, 47 times 0) [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]
- 263 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 7 times 0, 1, 11 times 0, 1, 15 times 0, 1, 20 times 0, 1, 25 times 0, 1, 32 times 0, 1, 38 times 0, 1, 43 times 0, 1, 47 times 0) [i] based on linear OA(3225, 2188, F3, 49) (dual of [2188, 1963, 50]-code), using
(194, 194+49, 317156)-Net in Base 3 — Upper bound on s
There is no (194, 243, 317157)-net in base 3, because
- 1 times m-reduction [i] would yield (194, 242, 317157)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 29 064958 506716 713130 731539 408351 807671 738408 887642 965924 014415 256264 505690 371027 025688 389984 020745 252883 998481 229089 > 3242 [i]