Best Known (168−110, 168, s)-Nets in Base 3
(168−110, 168, 48)-Net over F3 — Constructive and digital
Digital (58, 168, 48)-net over F3, using
- t-expansion [i] based on digital (45, 168, 48)-net over F3, using
- net from sequence [i] based on digital (45, 47)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 45 and N(F) ≥ 48, using
- net from sequence [i] based on digital (45, 47)-sequence over F3, using
(168−110, 168, 64)-Net over F3 — Digital
Digital (58, 168, 64)-net over F3, using
- t-expansion [i] based on digital (49, 168, 64)-net over F3, using
- net from sequence [i] based on digital (49, 63)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 49 and N(F) ≥ 64, using
- net from sequence [i] based on digital (49, 63)-sequence over F3, using
(168−110, 168, 211)-Net over F3 — Upper bound on s (digital)
There is no digital (58, 168, 212)-net over F3, because
- 2 times m-reduction [i] would yield digital (58, 166, 212)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3166, 212, F3, 108) (dual of [212, 46, 109]-code), but
- residual code [i] would yield OA(358, 103, S3, 36), but
- the linear programming bound shows that M ≥ 10156 978991 001331 162956 096121 250957 052835 549981 966253 500685 908151 917037 752938 447734 452120 610565 247729 783614 274683 456708 473505 647028 881750 777661 / 2 150100 658067 421186 781820 643305 293437 760326 405421 945649 628834 598190 514781 560026 381075 663150 228993 281922 344503 352960 > 358 [i]
- residual code [i] would yield OA(358, 103, S3, 36), but
- extracting embedded orthogonal array [i] would yield linear OA(3166, 212, F3, 108) (dual of [212, 46, 109]-code), but
(168−110, 168, 255)-Net in Base 3 — Upper bound on s
There is no (58, 168, 256)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 170 323088 348461 575979 709829 588714 133952 738594 471325 273502 873684 035332 682982 575105 > 3168 [i]