Best Known (194−125, 194, s)-Nets in Base 3
(194−125, 194, 48)-Net over F3 — Constructive and digital
Digital (69, 194, 48)-net over F3, using
- t-expansion [i] based on digital (45, 194, 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
(194−125, 194, 82)-Net over F3 — Digital
Digital (69, 194, 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
(194−125, 194, 271)-Net over F3 — Upper bound on s (digital)
There is no digital (69, 194, 272)-net over F3, because
- 2 times m-reduction [i] would yield digital (69, 192, 272)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3192, 272, F3, 123) (dual of [272, 80, 124]-code), but
- residual code [i] would yield OA(369, 148, S3, 41), but
- the linear programming bound shows that M ≥ 527 199306 719658 208896 926956 753282 382996 646971 832171 070963 054553 993200 450591 441546 901893 314193 819862 337393 820880 042121 319806 301584 641161 179874 156104 581689 649391 176134 052857 781324 366344 646605 508834 699899 215817 264311 251735 859253 836529 335378 412635 390318 558435 614367 704494 364599 306652 835549 960494 874188 110220 429716 297039 302052 791960 / 583209 910875 303263 469889 088835 657357 701840 865796 306372 080734 049489 546624 098606 841917 840472 812622 828337 614372 329030 815396 391551 616892 000268 241706 596845 254335 190377 639143 384747 649704 694395 992598 657828 113373 740022 140260 871631 828732 981772 576649 955489 025913 848348 858077 645730 817239 947263 609096 272537 > 369 [i]
- residual code [i] would yield OA(369, 148, S3, 41), but
- extracting embedded orthogonal array [i] would yield linear OA(3192, 272, F3, 123) (dual of [272, 80, 124]-code), but
(194−125, 194, 291)-Net in Base 3 — Upper bound on s
There is no (69, 194, 292)-net in base 3, because
- 16 times m-reduction [i] would yield (69, 178, 292)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3178, 292, S3, 109), but
- 8 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]
- 8 times code embedding in larger space [i] would yield OA(3186, 300, S3, 109), but
- extracting embedded orthogonal array [i] would yield OA(3178, 292, S3, 109), but