Best Known (73, 73+133, s)-Nets in Base 3
(73, 73+133, 48)-Net over F3 — Constructive and digital
Digital (73, 206, 48)-net over F3, using
- t-expansion [i] based on digital (45, 206, 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
(73, 73+133, 84)-Net over F3 — Digital
Digital (73, 206, 84)-net over F3, using
- t-expansion [i] based on digital (71, 206, 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
(73, 73+133, 272)-Net over F3 — Upper bound on s (digital)
There is no digital (73, 206, 273)-net over F3, because
- 1 times m-reduction [i] would yield digital (73, 205, 273)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3205, 273, F3, 132) (dual of [273, 68, 133]-code), but
- residual code [i] would yield OA(373, 140, S3, 44), but
- the linear programming bound shows that M ≥ 426 651196 533945 277794 177817 066459 131374 008202 085888 802659 036696 527669 202693 815983 584391 137459 056798 457669 192159 045403 286391 965778 304034 355162 563286 331591 445466 916914 887710 817006 739356 421510 777732 997222 018381 182300 894737 528738 685605 711329 173568 638990 281133 910218 248264 468221 824782 678390 971932 471290 987610 894145 690857 / 6226 806701 925935 069333 939189 184464 218829 831331 158728 544801 206473 585590 314042 486809 462072 681270 025131 928836 141769 329550 849992 179178 569640 566238 985428 411754 333557 519470 160622 103652 366214 816813 466703 978253 198332 731724 093345 562626 197678 209286 936852 140296 505811 857960 325584 621714 635265 > 373 [i]
- residual code [i] would yield OA(373, 140, S3, 44), but
- extracting embedded orthogonal array [i] would yield linear OA(3205, 273, F3, 132) (dual of [273, 68, 133]-code), but
(73, 73+133, 295)-Net in Base 3 — Upper bound on s
There is no (73, 206, 296)-net in base 3, because
- 24 times m-reduction [i] would yield (73, 182, 296)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3182, 296, S3, 109), but
- 4 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]
- 4 times code embedding in larger space [i] would yield OA(3186, 300, S3, 109), but
- extracting embedded orthogonal array [i] would yield OA(3182, 296, S3, 109), but