Best Known (81, ∞, s)-Nets in Base 27
(81, ∞, 324)-Net over F27 — Constructive and digital
Digital (81, m, 324)-net over F27 for arbitrarily large m, using
- net from sequence [i] based on digital (81, 323)-sequence over F27, using
- t-expansion [i] based on digital (72, 323)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 72 and N(F) ≥ 324, using
- F4 from the tower of function fields by Bezerra, GarcÃa, and Stichtenoth over F27 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 72 and N(F) ≥ 324, using
- t-expansion [i] based on digital (72, 323)-sequence over F27, using
(81, ∞, 325)-Net over F27 — Digital
Digital (81, m, 325)-net over F27 for arbitrarily large m, using
- net from sequence [i] based on digital (81, 324)-sequence over F27, using
- t-expansion [i] based on digital (48, 324)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 48 and N(F) ≥ 325, using
- t-expansion [i] based on digital (48, 324)-sequence over F27, using
(81, ∞, 2166)-Net in Base 27 — Upper bound on s
There is no (81, m, 2167)-net in base 27 for arbitrarily large m, because
- m-reduction [i] would yield (81, 4331, 2167)-net in base 27, but
- extracting embedded OOA [i] would yield OOA(274331, 2167, S27, 2, 4250), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 2016 733137 287115 960159 698738 619035 298540 238181 248101 133633 366182 711244 947227 784766 040892 047734 724930 830002 168821 918623 012046 038530 694109 645847 114666 567084 804309 930614 213972 140688 219925 264964 702220 934344 674107 855683 200883 954913 255109 363644 240551 878828 310935 964658 501226 371211 019512 633864 079291 873879 517367 687335 859898 048309 424867 111196 280107 805926 172379 954462 765678 816026 365077 932583 708304 685814 590865 201593 174167 132897 395316 058170 175573 439559 466541 310659 886935 550043 860738 011800 377778 791653 586687 909679 119779 052902 809744 396635 529884 120426 748333 590800 416921 088817 663018 527931 237896 056850 369393 199817 946177 234255 951171 671513 812837 334945 949595 081683 026412 920805 248153 203055 217753 385107 315970 471207 993726 758203 887146 850264 976938 383767 276599 310127 497223 024767 540711 760477 527175 349690 565986 064527 873217 851280 324300 048955 663621 925066 546421 118915 944849 929998 335968 544932 985969 226618 848167 172016 507419 508899 967106 979946 717415 757830 003241 237752 569918 894897 777449 879614 658588 951819 162271 168625 611081 996819 822195 921992 120714 818109 038520 402584 784347 002748 762011 577399 382721 731636 676801 696844 482402 153755 200832 468767 000806 905375 090491 984293 118885 331599 319235 412548 313130 514446 768426 534976 614964 852444 320628 597690 553297 581914 151744 217578 124694 483932 164551 167498 413540 250516 878714 678463 458463 786203 432527 055874 301454 857528 087219 186234 413040 668754 265336 627449 143302 358449 182163 365748 732007 746368 699915 906376 914823 834311 438810 577625 454342 419479 543114 486871 303058 218349 524468 441335 235249 828188 833465 388247 167269 034522 464136 549122 795125 875127 960892 205028 280919 914191 304158 043678 009127 294478 796076 917996 208867 530010 887394 838124 582647 181605 264109 382896 498332 453789 655904 394807 685232 706372 232388 358653 485881 372443 996654 662856 288107 407996 842743 970503 849750 884049 125514 555223 105041 049895 067544 091661 682625 664785 763359 509300 136049 235763 177014 988195 307245 089656 373949 960681 400816 740878 690136 143864 141226 730775 422540 957277 357723 639969 220275 446418 705020 129431 322177 118316 016129 370270 883328 163556 873475 563707 090558 693396 588115 863630 255453 322183 811108 511581 764132 601882 753686 313220 591228 004944 096354 274355 077030 667832 089471 200064 295426 537549 492614 239214 976123 949825 423232 326103 087000 913026 524449 157939 786772 127996 033862 914414 557108 560139 479637 142271 836138 077336 536697 772519 928620 198929 037010 969865 974709 804287 605607 855138 178654 833957 224404 496711 625198 499447 238414 697339 427732 152417 806737 066544 187847 822420 111502 396872 802241 444044 964557 659818 968163 309642 639798 480400 734829 434236 896894 636032 359852 241844 697245 262470 215884 822111 558618 347556 790467 335386 656942 720093 828789 926765 518181 132878 489762 999942 203014 410587 101718 383164 328980 243845 613370 011522 109718 152019 056089 006880 709505 811753 108235 934090 657371 015435 144453 761656 051928 208476 236626 905743 959848 764603 255061 052192 462229 790342 435566 092742 565882 911383 531992 983015 883843 823163 322532 770328 098320 908377 773724 219278 588211 045414 084331 960048 960557 876997 427631 973325 548717 740504 045364 643321 005168 383588 907402 776556 744619 238359 819289 032621 308371 648741 979675 125431 480289 469000 931989 470630 869772 165712 873482 332876 323235 497352 833923 800524 228403 824485 594766 944733 265509 289828 523323 532418 444951 524857 082353 379777 955896 636086 826277 963293 199919 189736 377928 874280 981334 585974 498016 353295 942555 088823 546673 320424 761530 746296 509795 399702 576333 884545 461267 305290 885762 025512 957729 925089 476310 298905 436466 521910 190759 504724 843335 754269 171646 305321 257612 933582 205320 891565 585564 881450 822101 397078 485934 769643 171424 178933 557754 928508 926646 461356 380020 699438 268605 630734 552958 547979 971127 214933 105327 678954 304875 801787 991939 910597 286014 237079 484163 547872 061073 588121 312344 139183 930899 568582 576664 275170 821360 767906 373167 492084 742480 931742 175005 939223 440165 187295 150140 789022 520281 873980 070170 700495 898387 685113 531013 612961 940450 204587 454283 848609 331323 747157 728262 676676 545329 554707 677294 902568 695134 122213 553391 237943 508623 623914 233347 838104 946704 095855 200097 356209 618972 533350 161700 745776 426247 494223 598684 079212 486190 665654 554217 829174 712786 571929 126242 744710 691435 754707 453816 222507 324931 691256 809669 416558 681171 418125 954172 235808 303095 125692 150138 544932 829668 268959 684938 386417 760881 803928 442107 321169 405825 715821 354911 017234 701846 182831 566863 294465 404849 878089 643030 774466 121217 546659 020182 270121 315063 915203 650322 590628 841407 156285 473372 879758 450865 947333 311468 056305 843169 153965 626934 091824 383382 442488 150957 971176 880628 176883 598035 238294 995027 482331 279630 711841 347101 152593 853976 249287 133561 371538 996061 087051 140728 032831 007862 199339 603520 959956 605209 586811 030409 528039 603133 000473 118818 229398 071082 875576 000212 611833 591202 867781 327286 765687 381425 656594 038504 710370 324965 679139 363828 136690 286292 847202 621306 793412 645143 767847 521047 861453 621844 781803 304022 042565 583709 834244 637310 579569 491160 135078 029682 737072 751164 267978 539167 033546 928650 499332 267963 545991 893280 445677 874999 492592 569950 532616 422358 974452 982105 847483 168529 706883 024999 257251 725887 971181 613782 705183 364116 894391 930713 146673 568110 306845 974152 558141 968306 388384 158920 750450 468069 722678 585252 466958 627563 369169 702520 208077 134682 623635 647590 463947 664340 567474 787119 243775 438645 539314 375082 084304 366647 374994 655574 383496 087567 611713 317083 725674 759351 819110 024283 876849 085566 133258 814937 882126 210839 745090 478925 163048 416101 278963 179541 177162 033470 946702 797839 174510 570854 960487 631144 323590 686552 534865 844163 358746 884997 809551 777647 611674 255241 656642 888375 380347 704706 987493 325514 166222 617214 868733 023442 548018 346275 117842 937321 261490 150164 895403 067945 270443 551966 302435 532019 004635 154868 551783 379491 318670 501495 764770 358973 091258 943752 706441 053946 967945 587494 899535 126677 023190 932107 003109 224616 156731 460254 013046 547577 834307 424977 803330 162328 503491 854849 077084 630322 851685 087107 227804 745972 675254 056017 979656 097063 314760 381264 752557 070418 015030 704575 990975 119161 529969 953343 089565 314834 990045 165637 219934 265197 408803 710786 663446 369865 346320 230258 163483 116614 792750 834875 477942 598092 141436 144338 530413 385234 195473 269416 754663 159810 874360 729349 782124 998899 670946 240568 201806 088417 477332 797114 830596 673147 857454 776055 435489 324845 849257 926001 214890 136107 905709 591679 432061 264427 516932 502210 969955 853608 896913 937510 279042 430900 812488 763597 742981 043771 886754 788737 673814 114317 591686 075879 114974 009757 501541 758859 850462 249119 724371 993663 095391 / 109 > 274331 [i]
- extracting embedded OOA [i] would yield OOA(274331, 2167, S27, 2, 4250), but