Best Known (85, ∞, s)-Nets in Base 27
(85, ∞, 324)-Net over F27 — Constructive and digital
Digital (85, m, 324)-net over F27 for arbitrarily large m, using
- net from sequence [i] based on digital (85, 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
(85, ∞, 325)-Net over F27 — Digital
Digital (85, m, 325)-net over F27 for arbitrarily large m, using
- net from sequence [i] based on digital (85, 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
(85, ∞, 2271)-Net in Base 27 — Upper bound on s
There is no (85, m, 2272)-net in base 27 for arbitrarily large m, because
- m-reduction [i] would yield (85, 4541, 2272)-net in base 27, but
- extracting embedded OOA [i] would yield OOA(274541, 2272, S27, 2, 4456), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 346556 556064 609268 835112 611730 657522 064199 966136 219018 533485 175678 519370 759468 968699 981897 086193 196766 401721 579136 250763 089769 630131 865489 326833 274523 914111 150536 336523 666769 468822 675251 444284 097965 221708 258860 562507 573406 800450 198965 394034 355602 413234 886454 900312 205149 657277 895036 584087 387323 978239 519771 433602 565156 084297 895762 168069 112468 137873 127196 736873 372993 352939 352621 318621 212199 809982 074648 874771 649460 106295 390474 809352 335874 371694 510574 973904 401664 473205 486628 445119 267775 879614 967909 222561 595935 066625 546741 339491 059390 895979 961299 685882 301181 418962 555818 220537 041916 406220 046477 162679 853946 443274 103551 969750 925342 734941 884604 857338 265219 991328 776820 900720 606807 797123 926089 581199 806581 388353 595428 662701 976942 941347 257049 694028 230713 228790 186552 173954 765925 533740 628757 327620 832368 655015 216277 546620 195050 000736 172010 295936 593039 335459 750548 551399 832310 230320 286275 738270 247273 822079 274665 281601 961612 056284 224769 214906 824810 431144 284860 440753 447882 243219 186123 024872 214711 589929 193181 732757 274505 729267 447089 992377 900842 908007 977918 831389 052795 517903 308216 199769 092283 691487 319013 181563 404810 503439 036221 713566 303881 626186 180620 156820 238259 698631 478149 856414 409377 609536 966684 604610 130744 264459 208663 004948 677518 555756 615462 244856 798726 537996 351387 929376 458758 443196 576849 603479 351440 718394 730182 533105 597393 050573 631005 273170 699990 822779 660442 854253 905243 958291 986471 634167 287161 613565 228047 099278 983563 959595 262051 439132 266936 634002 937587 449742 030218 270345 838726 703938 198250 394926 188983 059097 648727 723913 233075 680884 020496 426490 065411 560070 191583 233979 802251 006673 959014 387496 747644 980452 508419 784523 148701 542400 127544 836465 180959 528284 334948 560072 639957 998175 790262 157945 801175 031464 178309 395502 948953 882385 121652 454732 262637 627430 301458 933431 671934 318617 477688 489943 829202 122535 284791 120054 365337 197843 971522 637453 154592 186696 032522 147773 010992 115123 026848 599502 193797 403932 790781 750431 689189 380925 940063 544535 104240 149703 750813 942892 225897 524407 499038 339423 558705 879394 315366 875103 971661 482271 162274 219861 965490 282205 390491 376399 367280 620532 107700 857482 606337 542816 375745 916392 059429 317923 531452 369125 774868 810583 512032 183707 874615 794275 997887 541733 164405 238868 849561 659385 645805 558133 988764 715419 847843 088364 087572 604049 580169 982582 140718 375005 816864 090493 809681 249355 959431 390245 161130 773793 833999 583267 476397 214924 647826 393433 051904 547505 190444 957062 774125 006523 339720 072427 837513 680558 879344 349937 695084 995301 962624 169228 666907 960053 505870 880457 393916 135338 757592 600759 537898 482882 500812 478321 120493 886219 119292 844529 189565 331090 003694 081998 824742 427468 788837 185511 125900 942957 420985 256998 381173 142623 861213 449414 116684 361239 970639 383011 060406 897912 594150 684186 529541 222366 395493 896348 509724 042450 204532 680620 121347 787565 559127 873440 260340 056796 756893 732050 714195 432391 403610 452733 493624 283246 228972 701622 866571 044080 141278 693630 128408 707671 035351 264956 646585 080671 922884 826320 546651 603121 704707 888243 001279 101940 153725 001495 720131 023661 219857 727387 606310 865596 141624 824776 612330 723201 514188 382854 737037 060801 237964 726321 343774 472187 857910 075986 465589 483024 627164 573260 882816 962367 860159 853741 572259 330706 720298 482350 065028 427948 363457 664998 355188 562100 060596 223434 580585 835656 670538 298362 674206 821670 830169 350459 618437 603380 733718 210898 058571 009483 977150 661306 173292 005200 085811 588851 819845 977387 020768 272340 889398 903754 573438 414360 620927 154665 358882 241494 940490 367729 669909 433806 406427 048367 948509 211904 721316 424159 974852 950489 185617 312841 385522 675511 130782 795922 649214 722450 052071 817246 851964 010675 346889 412553 374014 981183 797495 359280 852295 564293 960673 852719 676102 791485 634186 482349 096778 890937 547543 032229 662230 448474 045227 686289 188199 184927 570393 521579 789357 925941 297600 090305 595841 473683 270461 259404 937264 975624 000120 065862 849253 997993 501844 631661 282717 509744 645457 076037 345667 800344 939716 376568 970503 080573 871062 940890 531028 101672 005454 690274 829917 638647 325678 835991 516858 053410 565049 891166 379915 295928 589970 401118 363591 424370 352566 045835 089076 843113 283484 639322 456758 521099 303734 811393 434786 905677 033038 244166 943554 863414 961997 123350 265968 487956 149468 138989 252389 542968 620146 834188 837401 448791 360874 905271 821820 813758 343635 129245 309191 335355 464499 915585 748620 578005 394211 148809 219770 298587 046428 532561 533928 571296 967227 885080 841420 087577 107309 512817 494089 337515 322714 138890 254081 196142 993763 683286 078042 582012 136694 544015 195786 992824 107602 707173 811148 004034 854349 116626 223064 592017 720820 193818 031774 514943 981580 763806 155088 592368 259181 656311 440705 831476 738061 629170 628787 163568 148469 702233 262018 171941 596866 409120 119944 763742 738014 039063 083862 178133 738183 984161 411931 437045 088546 716959 616758 909992 120101 318240 924729 478994 935622 757692 487026 644939 984390 923163 564834 578108 473978 777027 006783 843409 770342 435048 944794 689038 124370 348337 102271 302073 508174 457935 890462 464712 691877 037119 586889 610855 121369 795922 451545 159612 261991 207384 416151 752866 388586 987604 536201 236662 764475 737866 739912 176110 007256 002567 777122 234299 357519 225381 446670 813984 104667 722946 601552 017422 215353 153148 109505 521233 078775 969459 267044 794426 929599 110091 526404 156012 743888 019526 445536 766795 328397 552553 167996 229022 916580 880541 441924 190082 975977 461131 546173 929840 114274 228230 964506 750570 782910 876776 800004 569082 278633 250775 736504 362331 985620 254024 683716 523749 121919 982196 596947 642705 841016 510672 006670 401089 284169 375751 337529 207160 368791 533279 646627 979666 301965 490064 783892 047963 588679 075576 770091 893822 707817 893718 030801 048175 377236 574552 323593 002199 072829 897191 540204 045248 802798 840171 212212 714271 111631 835982 707794 314998 505311 032295 433258 791972 150254 881941 830314 441827 242614 518058 658847 016438 225527 089297 779287 705668 081048 299078 485524 096043 961969 816190 529149 958529 562541 406373 550768 572546 202697 295364 150262 839553 629228 674606 472571 196587 915469 094305 636223 945793 947324 829154 746498 103570 385399 154700 033597 250652 228681 729592 489419 749536 073392 850932 161310 390715 966267 260883 014621 998636 059057 243132 656221 420134 936419 588936 115341 643430 472509 920042 129767 462605 265014 787463 530453 885295 838243 526895 239232 649936 629144 776664 624405 286238 972994 086662 320110 138753 780789 398564 463930 345506 110634 244600 540780 355182 802041 895971 019695 981611 069497 851798 257885 585251 677859 308499 633828 990229 385391 131122 722029 690285 150069 746267 422863 696698 777323 360050 577014 722485 320092 524502 440667 119976 002239 284989 109874 560589 673767 616015 004631 176119 952760 413354 146040 859463 639972 741636 568601 034376 749834 980604 690005 034603 842010 935612 199889 407741 935675 549149 768310 661262 501058 043777 189383 329157 308222 866824 728080 835018 689258 754031 044435 891497 / 4457 > 274541 [i]
- extracting embedded OOA [i] would yield OOA(274541, 2272, S27, 2, 4456), but