Best Known (198−126, 198, s)-Nets in Base 3
(198−126, 198, 48)-Net over F3 — Constructive and digital
Digital (72, 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
(198−126, 198, 84)-Net over F3 — Digital
Digital (72, 198, 84)-net over F3, using
- t-expansion [i] based on digital (71, 198, 84)-net over F3, using
- net from sequence [i] based on digital (71, 83)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 71 and N(F) ≥ 84, using
- net from sequence [i] based on digital (71, 83)-sequence over F3, using
(198−126, 198, 292)-Net over F3 — Upper bound on s (digital)
There is no digital (72, 198, 293)-net over F3, because
- extracting embedded orthogonal array [i] would yield linear OA(3198, 293, F3, 126) (dual of [293, 95, 127]-code), but
- residual code [i] would yield OA(372, 166, S3, 42), but
- the linear programming bound shows that M ≥ 3708 177920 981865 373588 437755 619329 963493 853633 501053 221945 962924 753226 122178 646596 684570 785490 813971 229149 075678 252631 823220 208670 888172 933548 774937 964032 317385 185116 199333 762378 663726 021334 995353 647735 985557 719209 619572 304083 217640 938540 673734 696000 / 156561 852463 373651 529187 845061 951139 819618 179813 617086 729231 310074 778337 668217 120160 348367 793511 265448 893510 422713 110540 923424 266826 849685 171967 550061 800650 508114 845380 137520 223684 048212 787473 386711 548511 501577 011613 > 372 [i]
- residual code [i] would yield OA(372, 166, S3, 42), but
(198−126, 198, 294)-Net in Base 3 — Upper bound on s
There is no (72, 198, 295)-net in base 3, because
- 17 times m-reduction [i] would yield (72, 181, 295)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3181, 295, S3, 109), but
- 5 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]
- 5 times code embedding in larger space [i] would yield OA(3186, 300, S3, 109), but
- extracting embedded orthogonal array [i] would yield OA(3181, 295, S3, 109), but