Best Known (61, 61+107, s)-Nets in Base 3
(61, 61+107, 48)-Net over F3 — Constructive and digital
Digital (61, 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
(61, 61+107, 64)-Net over F3 — Digital
Digital (61, 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
(61, 61+107, 265)-Net over F3 — Upper bound on s (digital)
There is no digital (61, 168, 266)-net over F3, because
- 2 times m-reduction [i] would yield digital (61, 166, 266)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3166, 266, F3, 105) (dual of [266, 100, 106]-code), but
- residual code [i] would yield OA(361, 160, S3, 35), but
- the linear programming bound shows that M ≥ 1093 910277 777742 998768 653454 514500 328731 948244 229348 565648 421721 114272 536195 / 8588 577766 921583 861591 295968 393699 452287 752003 > 361 [i]
- residual code [i] would yield OA(361, 160, S3, 35), but
- extracting embedded orthogonal array [i] would yield linear OA(3166, 266, F3, 105) (dual of [266, 100, 106]-code), but
(61, 61+107, 278)-Net in Base 3 — Upper bound on s
There is no (61, 168, 279)-net in base 3, because
- 1 times m-reduction [i] would yield (61, 167, 279)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 48 357369 921560 064593 831736 412548 146028 821144 725437 577521 235635 072491 361032 421255 > 3167 [i]