Best Known (189−122, 189, s)-Nets in Base 3
(189−122, 189, 48)-Net over F3 — Constructive and digital
Digital (67, 189, 48)-net over F3, using
- t-expansion [i] based on digital (45, 189, 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
(189−122, 189, 72)-Net over F3 — Digital
Digital (67, 189, 72)-net over F3, using
- net from sequence [i] based on digital (67, 71)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 67 and N(F) ≥ 72, using
(189−122, 189, 261)-Net over F3 — Upper bound on s (digital)
There is no digital (67, 189, 262)-net over F3, because
- 2 times m-reduction [i] would yield digital (67, 187, 262)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3187, 262, F3, 120) (dual of [262, 75, 121]-code), but
- residual code [i] would yield OA(367, 141, S3, 40), but
- the linear programming bound shows that M ≥ 37 867593 653082 242764 820378 309568 512330 272405 522468 546682 102535 276945 315487 241500 791718 572671 989136 089217 024902 800421 918620 245031 068490 122189 898639 027163 073470 583829 048249 943822 337971 643956 565391 295309 197864 197892 549692 269713 467891 498765 676112 582800 067831 676520 523187 534107 797030 794015 010412 084605 290056 749825 / 403083 479476 744301 411084 077038 795612 098817 078378 104739 489410 795719 130624 349638 943754 745948 064072 884391 693439 403958 404030 714901 482333 773784 235914 986441 493247 898381 831927 994462 420009 556149 986521 269439 451417 628689 899317 187610 402011 435909 979065 730186 126000 512248 330299 309348 182161 > 367 [i]
- residual code [i] would yield OA(367, 141, S3, 40), but
- extracting embedded orthogonal array [i] would yield linear OA(3187, 262, F3, 120) (dual of [262, 75, 121]-code), but
(189−122, 189, 289)-Net in Base 3 — Upper bound on s
There is no (67, 189, 290)-net in base 3, because
- 13 times m-reduction [i] would yield (67, 176, 290)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3176, 290, S3, 109), but
- 10 times code embedding in larger space [i] would yield OA(3186, 300, S3, 109), but
- the linear programming bound shows that M ≥ 744 646766 850603 526568 601268 797422 932026 535817 992617 489413 629925 481692 835741 945421 210090 969950 381274 252816 504020 353660 158683 457200 925516 493661 440389 831805 338854 508258 811020 990136 433923 762970 235494 388055 889637 880458 633459 997426 817105 857222 614437 226587 / 10512 885684 310613 723066 595609 767913 116130 406095 478720 857818 594175 294259 025889 504412 156455 324830 169263 105843 888569 812652 562285 873819 443701 107860 316189 843301 339995 > 3186 [i]
- 10 times code embedding in larger space [i] would yield OA(3186, 300, S3, 109), but
- extracting embedded orthogonal array [i] would yield OA(3176, 290, S3, 109), but