Best Known (71, 71+128, s)-Nets in Base 3
(71, 71+128, 48)-Net over F3 — Constructive and digital
Digital (71, 199, 48)-net over F3, using
- t-expansion [i] based on digital (45, 199, 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
(71, 71+128, 84)-Net over F3 — Digital
Digital (71, 199, 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
(71, 71+128, 279)-Net over F3 — Upper bound on s (digital)
There is no digital (71, 199, 280)-net over F3, because
- 2 times m-reduction [i] would yield digital (71, 197, 280)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3197, 280, F3, 126) (dual of [280, 83, 127]-code), but
- residual code [i] would yield OA(371, 153, S3, 42), but
- the linear programming bound shows that M ≥ 1 055633 583684 412124 576484 170255 131218 780961 470521 927693 463389 568320 189199 289559 169330 933131 798207 551957 567481 144539 850575 395329 102764 159939 358868 077128 618402 122131 171260 366925 398215 527745 180439 670326 837896 090390 666310 679248 321377 576820 431328 275321 422069 606660 949118 551478 240579 301418 914961 095796 891840 / 138 376916 366765 245149 784082 130406 254967 569712 303599 812312 770276 538679 404780 518662 697999 679212 679952 906967 837259 967292 004231 340859 946531 558376 639875 802868 726402 799726 504048 508110 636251 941347 913705 438595 007008 728154 695053 276605 319416 010222 452132 016051 115790 666606 828591 > 371 [i]
- residual code [i] would yield OA(371, 153, S3, 42), but
- extracting embedded orthogonal array [i] would yield linear OA(3197, 280, F3, 126) (dual of [280, 83, 127]-code), but
(71, 71+128, 293)-Net in Base 3 — Upper bound on s
There is no (71, 199, 294)-net in base 3, because
- 19 times m-reduction [i] would yield (71, 180, 294)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3180, 294, S3, 109), but
- 6 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]
- 6 times code embedding in larger space [i] would yield OA(3186, 300, S3, 109), but
- extracting embedded orthogonal array [i] would yield OA(3180, 294, S3, 109), but