Best Known (37, ∞, s)-Nets in Base 64
(37, ∞, 513)-Net over F64 — Constructive and digital
Digital (37, m, 513)-net over F64 for arbitrarily large m, using
- net from sequence [i] based on digital (37, 512)-sequence over F64, using
- t-expansion [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- t-expansion [i] based on digital (28, 512)-sequence over F64, using
(37, ∞, 540)-Net over F64 — Digital
Digital (37, m, 540)-net over F64 for arbitrarily large m, using
- net from sequence [i] based on digital (37, 539)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 37 and N(F) ≥ 540, using
(37, ∞, 2458)-Net in Base 64 — Upper bound on s
There is no (37, m, 2459)-net in base 64 for arbitrarily large m, because
- m-reduction [i] would yield (37, 4915, 2459)-net in base 64, but
- extracting embedded OOA [i] would yield OOA(644915, 2459, S64, 2, 4878), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 1 971040 087220 852318 442733 170282 243303 749419 521384 077473 666810 901713 411250 588778 440847 630249 168498 381499 086042 438079 245648 921234 362528 791973 485726 139779 423348 849307 316077 100736 333508 592167 568204 036631 021209 087476 456011 424121 319262 898753 865615 232861 485269 200494 558855 847271 659821 761330 982873 364270 043004 381348 988217 899899 796716 881889 534236 155703 647345 497096 209540 231717 521603 620045 383336 337486 166632 677259 207702 579041 922626 460178 765213 581010 485365 480002 769448 858380 765391 205133 025704 800009 501931 436543 464960 628651 815114 106506 395765 545489 426998 835560 778213 413481 626005 799220 726432 580693 592631 187055 159675 983935 860717 096024 754863 152921 751655 336291 944647 836007 606737 809776 738686 712002 162702 118058 065876 500225 206658 861291 485091 094393 430020 703333 102645 018913 688861 968047 428491 100370 102660 888102 195014 608156 665255 738444 547374 676864 646297 109178 304384 148209 565537 435891 565196 282341 921312 421102 961208 777542 595308 931381 361254 081506 717793 465032 493802 716713 832645 252307 176551 907071 836697 075625 380915 955121 424318 709200 539128 131585 405958 913104 644116 967032 204515 220483 908376 832269 381584 285142 780950 832110 907761 063965 832539 367753 704851 886657 746621 427908 511686 987236 190004 512892 862333 836115 344927 825889 061696 224186 345001 842932 146532 394754 634750 150155 545280 247994 961790 652083 855962 282932 894356 384738 771699 709404 847982 414754 197438 886744 273077 603105 141903 335879 677924 624942 965349 164551 542769 494321 842155 263922 710400 417091 411608 077364 953009 247750 839971 678823 420036 378648 826102 407759 099010 870406 400912 957057 041695 432903 149825 231167 582619 533117 553348 303514 891593 363719 063805 856831 621734 672594 193012 106678 407864 862821 490553 237073 262354 584292 324659 227469 120232 476295 506202 849795 243845 031816 498204 676183 565850 202804 853414 375700 010025 113858 799402 357533 469532 724012 374977 872938 970217 664119 120174 262552 056490 751633 371937 791799 610823 665036 056467 640909 548413 795578 041536 464701 200873 083956 354117 216121 596050 849074 794472 561772 912947 578278 842747 236137 284147 331026 630981 870452 973130 634105 892083 462574 084561 217526 556383 442941 611992 414982 104570 461447 157746 638650 120779 781859 982246 370008 075090 407650 509228 992022 579616 756071 441915 515683 560005 284337 289121 287131 349174 798546 141333 281346 635359 599916 377166 736752 668200 564534 716036 846421 264918 009118 376913 151615 779165 073000 072457 657539 595107 543380 438920 577575 739164 210730 044965 525739 612225 238303 364035 117442 996284 045174 229251 416355 232255 872405 345874 625147 145644 343821 251706 279933 842216 237193 000150 333299 991760 388936 350305 259719 206531 994316 295080 795227 211327 427780 180066 677491 411220 680816 042727 363660 190287 071600 549119 878556 124797 762624 159220 843217 047641 457516 812335 728179 778744 099636 655134 894181 539294 259655 394647 722484 457828 580464 253245 450807 515218 582904 113565 256449 344426 002633 236601 253453 954885 099451 230446 673797 472720 981534 200428 343634 320742 872930 449083 768575 584197 205137 357896 703400 165763 790153 555508 863204 753478 888029 881359 775387 588668 450364 147354 007053 160086 681909 447370 565841 636452 493008 296550 407508 295386 017768 417971 387764 777250 909985 148999 479351 308785 584539 135756 703375 488260 875478 324749 610634 240129 112351 670804 258450 940438 319363 668961 938361 067819 690341 379495 561898 987003 622834 918253 695680 894976 252558 050236 407727 228893 616094 233120 901097 799621 814526 081605 410631 152140 442072 914543 593703 609918 669068 184780 552665 475374 617190 538092 725856 114319 663782 283325 258081 909876 507813 679523 160413 982981 732893 279373 129462 757657 425312 890653 582170 863425 723501 976091 771318 852238 790206 262449 068979 224906 038313 817503 033718 133304 538029 420269 949907 361828 650124 096359 482697 698175 948130 659198 314098 635442 969807 399803 529401 473211 369734 325023 863320 887736 369841 273377 966950 234570 868066 085985 951091 991173 157412 268616 252289 345373 903603 363661 868860 290152 947660 601536 629385 867932 746359 569897 467906 942610 324022 908928 956907 267960 146498 563604 689434 438882 937377 109142 548086 547270 199022 455580 319005 300961 687859 890463 493490 371559 467481 290890 920241 131801 359234 153673 207552 285998 357624 477450 699297 179497 501546 272622 688397 238417 374513 921799 915986 410801 202172 561383 375470 817555 857353 208278 909941 118939 276016 948957 070252 839582 748118 939039 188447 587235 923270 539555 081907 495606 816835 232451 021578 560827 396856 978135 654559 294105 774446 404354 049658 340018 936093 741688 197360 365390 390893 344119 700894 553726 359861 973586 876321 594265 763610 031978 556112 286571 033219 451170 483020 334513 982699 823572 250208 827372 257496 961739 491506 719645 176615 318304 225408 264737 601236 763132 340181 739606 237540 562827 040770 690853 064215 221503 862633 718905 263543 447249 024311 888773 417942 393153 608289 581058 564814 404113 878376 011917 658518 502619 893267 913289 242035 334302 785689 369885 334986 605884 720590 689178 079610 749474 416286 928292 529489 999025 877964 783353 664714 997912 689400 078982 907451 901426 735592 310830 806934 035022 469657 841290 979657 535989 904537 051570 368393 865041 394334 466029 816364 011214 209842 249149 042941 134381 529331 847787 012012 579072 216495 529369 263328 060169 099701 838672 246565 514261 719279 847519 435725 033118 553369 300926 515175 668812 097646 907629 802940 401150 296524 460031 581439 223498 891471 972453 494519 396452 484509 629315 240095 780406 611947 893838 378288 149736 153901 996482 311440 114355 182471 933459 057760 461765 641445 238014 623420 183440 346040 662194 975373 403915 355610 113037 644593 664710 627630 614797 116292 675468 575569 024995 758848 615648 402275 099734 639436 214890 193057 555950 081040 154092 900347 703976 430281 351072 701821 293158 449898 636451 661638 140337 883421 824732 366074 351764 223290 550069 237975 493860 376172 647414 248129 948839 877803 872079 091727 232091 485554 886758 680646 539376 252492 909348 773196 250962 652193 232885 333278 979066 511444 685323 014490 895119 383771 060059 259058 519799 783721 455820 570819 858358 975962 218150 643229 731263 611294 880781 575771 715728 402882 102190 374270 998709 562370 919933 226223 220384 521105 359545 036303 363923 458670 566797 894330 002930 456097 856030 140530 061230 244348 494636 608151 061565 775135 443111 907227 288515 032173 255296 257507 357047 348522 522681 964335 862947 771272 089379 793597 762556 163741 379305 366183 209922 441792 845346 050253 138707 510060 147211 027061 744005 913262 917518 841310 984107 673069 868223 832573 337065 960252 460339 394099 795324 037162 126635 225551 514807 706258 204466 027884 343964 005956 832544 827380 209599 285386 436945 861965 593591 600344 067636 493087 535897 053524 868231 632197 876616 140674 001502 418823 357728 770055 908928 568201 853488 943011 769015 630312 811797 733301 136610 758711 348171 889726 981124 907834 638867 462242 377765 393043 101908 130248 839387 165155 289169 779141 281658 853637 609332 085630 210998 155517 252204 954020 927353 943029 553497 724220 531627 334252 702118 215296 741801 648142 488471 542527 779162 679504 819359 439824 744561 469511 339935 913287 538384 442455 992710 007178 273714 259938 743856 075268 255448 263851 754529 380728 976166 689462 377758 894523 582355 277299 609558 196487 961748 727470 526051 529041 309606 458369 752196 749875 490502 771215 411750 837137 687498 386297 557689 953336 481844 412514 870207 107253 407751 119751 329441 951961 851467 974837 307100 517622 878676 327025 527840 443212 254025 250364 147620 953163 692572 493174 294014 320695 620495 353400 382536 671005 804059 351453 076839 693537 518296 570603 366024 941635 959569 584870 849913 409586 038798 442197 903737 976389 172835 865171 426809 789249 293695 690193 953684 667932 267632 037016 782102 961875 032481 619078 839716 533282 076294 625685 051634 609286 777665 607374 238820 805589 583508 347507 149213 980240 997089 854197 380800 536628 559944 842239 314838 960805 222285 129073 749392 893103 145536 658501 296415 311294 707029 368539 661710 541777 137594 490209 212098 146057 941500 563221 522799 051411 261686 704259 526943 370935 030869 274971 291797 237374 397177 648959 093260 770094 945491 782898 316808 220673 125720 253069 689162 481708 877012 628928 916062 625655 957341 160684 450146 486816 587849 813828 750912 736074 918974 129670 639857 032139 794399 630898 843059 377704 180923 752367 676422 617864 564726 082960 292094 166782 236203 583159 787260 924078 383783 995703 587178 595105 971152 448980 492894 472920 255634 328215 264360 064660 669117 588583 167523 967851 976007 801464 965830 749273 357406 809976 664004 683098 531142 721195 409574 223245 215499 741329 330523 538599 768634 336767 803970 568936 240108 124186 145980 947114 119590 957079 634024 853191 264322 520971 146589 138309 856092 678766 411349 709061 677414 995694 465752 357387 391378 041124 842024 105800 730352 452979 338551 402433 815182 723113 976785 861009 373345 475148 623386 400722 504576 087763 540446 224454 725813 327812 108927 814527 159123 384478 074118 762340 273566 700193 333616 407643 951961 568538 982944 929445 371604 329101 795877 214309 942906 745535 726993 905894 085930 450434 132227 816609 260201 225781 434513 294151 167118 636219 517951 749855 602923 936059 400319 861223 793956 945785 649393 159764 420974 665144 987801 507000 227582 590296 850532 771085 961743 171882 773313 035596 141970 705637 945685 006941 513949 972066 434082 673564 165696 461707 122467 551553 681323 082642 111018 956472 040024 783763 222565 728706 657925 731056 544443 475048 530624 685077 790314 204963 986981 157269 206784 161039 506795 546288 974815 305933 564912 782994 045057 531978 540059 070647 439161 173224 223993 803494 780686 783094 201325 205138 858445 328220 065255 503456 393305 116916 203796 733600 117489 266702 670820 266103 631551 060058 520982 388418 200227 525600 337006 214874 332197 108895 456356 169063 312260 179582 242792 884029 440117 922311 106929 533539 896961 008871 229553 058599 689974 232118 438113 422561 586711 081273 786368 / 697 > 644915 [i]
- extracting embedded OOA [i] would yield OOA(644915, 2459, S64, 2, 4878), but