Best Known (198−128, 198, s)-Nets in Base 3
(198−128, 198, 48)-Net over F3 — Constructive and digital
Digital (70, 198, 48)-net over F3, using
- t-expansion [i] based on digital (45, 198, 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
(198−128, 198, 82)-Net over F3 — Digital
Digital (70, 198, 82)-net over F3, using
- t-expansion [i] based on digital (69, 198, 82)-net over F3, using
- net from sequence [i] based on digital (69, 81)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 69 and N(F) ≥ 82, using
- net from sequence [i] based on digital (69, 81)-sequence over F3, using
(198−128, 198, 267)-Net over F3 — Upper bound on s (digital)
There is no digital (70, 198, 268)-net over F3, because
- 2 times m-reduction [i] would yield digital (70, 196, 268)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3196, 268, F3, 126) (dual of [268, 72, 127]-code), but
- residual code [i] would yield OA(370, 141, S3, 42), but
- the linear programming bound shows that M ≥ 4277 565698 839384 010170 589716 887095 058169 368020 564397 199414 212109 971383 967143 580539 372570 911510 043347 952663 370579 446414 731408 702577 108148 457502 649267 485100 181078 723724 847585 165364 997228 804035 089921 869721 103476 315522 380830 764656 460391 496415 038990 939869 773410 036939 045606 959626 591763 297984 748711 096468 331745 518174 622445 985981 095385 / 1 551512 830602 858860 338181 158623 054002 886482 377025 245272 895207 545358 537220 058041 448632 273527 418957 014308 934517 283125 135524 500935 334910 128739 898590 747907 120072 589632 997168 116604 990877 528573 097744 651010 819159 741742 497065 834837 245926 336361 938850 088121 503037 789061 126850 003819 367246 912997 850917 454263 > 370 [i]
- residual code [i] would yield OA(370, 141, S3, 42), but
- extracting embedded orthogonal array [i] would yield linear OA(3196, 268, F3, 126) (dual of [268, 72, 127]-code), but
(198−128, 198, 292)-Net in Base 3 — Upper bound on s
There is no (70, 198, 293)-net in base 3, because
- 19 times m-reduction [i] would yield (70, 179, 293)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3179, 293, S3, 109), but
- 7 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]
- 7 times code embedding in larger space [i] would yield OA(3186, 300, S3, 109), but
- extracting embedded orthogonal array [i] would yield OA(3179, 293, S3, 109), but