Best Known (69, 69+126, s)-Nets in Base 3
(69, 69+126, 48)-Net over F3 — Constructive and digital
Digital (69, 195, 48)-net over F3, using
- t-expansion [i] based on digital (45, 195, 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
(69, 69+126, 82)-Net over F3 — Digital
Digital (69, 195, 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
(69, 69+126, 254)-Net over F3 — Upper bound on s (digital)
There is no digital (69, 195, 255)-net over F3, because
- extracting embedded orthogonal array [i] would yield linear OA(3195, 255, F3, 126) (dual of [255, 60, 127]-code), but
- residual code [i] would yield OA(369, 128, S3, 42), but
- the linear programming bound shows that M ≥ 608 001198 593304 106218 907830 686907 775076 916565 165378 554092 710276 609555 357826 501088 330911 962882 720034 814817 891444 682681 838596 665067 963409 592743 684436 157686 057509 559170 713241 809051 496946 469627 269569 367596 452422 964732 120362 943819 537136 492537 / 700063 514007 690288 493623 500405 031069 826547 171422 379027 234454 614217 062891 744996 343198 204037 257328 431813 156380 977077 721464 526397 768944 327663 331813 298588 235435 706684 439261 947914 591660 379569 272018 701709 945865 > 369 [i]
- residual code [i] would yield OA(369, 128, S3, 42), but
(69, 69+126, 291)-Net in Base 3 — Upper bound on s
There is no (69, 195, 292)-net in base 3, because
- 17 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