Best Known (71, 71+129, s)-Nets in Base 3
(71, 71+129, 48)-Net over F3 — Constructive and digital
Digital (71, 200, 48)-net over F3, using
- t-expansion [i] based on digital (45, 200, 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+129, 84)-Net over F3 — Digital
Digital (71, 200, 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+129, 264)-Net over F3 — Upper bound on s (digital)
There is no digital (71, 200, 265)-net over F3, because
- extracting embedded orthogonal array [i] would yield linear OA(3200, 265, F3, 129) (dual of [265, 65, 130]-code), but
- residual code [i] would yield OA(371, 135, S3, 43), but
- the linear programming bound shows that M ≥ 15517 693217 689196 948814 275840 217839 188528 833666 506027 197482 862926 479145 955220 202759 225092 599101 677573 597273 739795 383261 969362 675805 202325 361940 553286 399020 268827 436811 157921 842316 026009 799441 891519 060953 199556 255550 386399 744243 197229 021375 769233 758290 438783 812037 692310 974622 568682 535976 282881 / 1 978655 878974 329766 696946 944041 110095 992060 424182 878846 382180 050085 637907 736643 328890 021267 752512 900990 168702 902588 902931 052866 529048 536876 574704 276924 879749 598186 293521 925859 415088 535294 830297 315800 130604 056048 836627 827058 062734 488069 759928 856867 528845 559020 > 371 [i]
- residual code [i] would yield OA(371, 135, S3, 43), but
(71, 71+129, 293)-Net in Base 3 — Upper bound on s
There is no (71, 200, 294)-net in base 3, because
- 20 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