Best Known (182−118, 182, s)-Nets in Base 3
(182−118, 182, 48)-Net over F3 — Constructive and digital
Digital (64, 182, 48)-net over F3, using
- t-expansion [i] based on digital (45, 182, 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
(182−118, 182, 64)-Net over F3 — Digital
Digital (64, 182, 64)-net over F3, using
- t-expansion [i] based on digital (49, 182, 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
(182−118, 182, 240)-Net over F3 — Upper bound on s (digital)
There is no digital (64, 182, 241)-net over F3, because
- 1 times m-reduction [i] would yield digital (64, 181, 241)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3181, 241, F3, 117) (dual of [241, 60, 118]-code), but
- residual code [i] would yield OA(364, 123, S3, 39), but
- the linear programming bound shows that M ≥ 3 296455 091967 658523 249507 296896 666417 309987 180746 959421 449762 980285 133004 936802 465483 658008 896383 778374 990127 915613 560505 830406 115220 454221 102979 294779 354315 375193 344486 780900 933129 866677 553427 395911 625133 984285 906366 542683 097491 785388 276692 273280 005567 961514 569303 / 895740 871596 337029 456022 571186 881299 264548 635548 913914 496610 386821 206881 792534 185298 576348 098257 958155 929354 264087 974570 822791 463426 021561 519213 167582 882120 884894 537569 804102 970266 339725 717861 197193 058544 988821 955797 835070 338737 147153 > 364 [i]
- residual code [i] would yield OA(364, 123, S3, 39), but
- extracting embedded orthogonal array [i] would yield linear OA(3181, 241, F3, 117) (dual of [241, 60, 118]-code), but
(182−118, 182, 283)-Net in Base 3 — Upper bound on s
There is no (64, 182, 284)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 747 854026 255698 212529 896075 289134 428224 340023 589284 123447 310172 495535 974621 302251 813585 > 3182 [i]