Best Known (72, 72+130, s)-Nets in Base 3
(72, 72+130, 48)-Net over F3 — Constructive and digital
Digital (72, 202, 48)-net over F3, using
- t-expansion [i] based on digital (45, 202, 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
(72, 72+130, 84)-Net over F3 — Digital
Digital (72, 202, 84)-net over F3, using
- t-expansion [i] based on digital (71, 202, 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
(72, 72+130, 277)-Net over F3 — Upper bound on s (digital)
There is no digital (72, 202, 278)-net over F3, because
- 1 times m-reduction [i] would yield digital (72, 201, 278)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3201, 278, F3, 129) (dual of [278, 77, 130]-code), but
- residual code [i] would yield OA(372, 148, S3, 43), but
- the linear programming bound shows that M ≥ 140616 689849 151177 540174 469107 623214 902524 373234 563713 296619 681304 722457 754991 147093 951918 153451 548407 544582 982455 263586 804453 976984 329877 768679 866003 344857 085709 968192 295150 158405 086355 117293 815995 399786 105620 103676 565115 407129 402845 238362 530278 975728 254191 863354 567751 991978 849547 640077 469027 501832 766234 432494 073485 807433 390139 517780 408495 / 5 598581 050097 351428 044464 209595 565234 397823 540732 261512 489572 263757 723078 348205 051059 822908 682186 213397 624116 594971 139254 976893 145418 485319 486609 248744 469897 672953 851007 182218 513650 832206 934424 052154 431313 143245 320083 612964 606743 062220 611961 594548 775648 983859 483908 360052 607131 075364 504891 427518 044319 045951 > 372 [i]
- residual code [i] would yield OA(372, 148, S3, 43), but
- extracting embedded orthogonal array [i] would yield linear OA(3201, 278, F3, 129) (dual of [278, 77, 130]-code), but
(72, 72+130, 294)-Net in Base 3 — Upper bound on s
There is no (72, 202, 295)-net in base 3, because
- 21 times m-reduction [i] would yield (72, 181, 295)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3181, 295, S3, 109), but
- 5 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]
- 5 times code embedding in larger space [i] would yield OA(3186, 300, S3, 109), but
- extracting embedded orthogonal array [i] would yield OA(3181, 295, S3, 109), but