MinT - Best Known (113, ∞, s)-Nets in Base 16

Best Known (113, ∞, s)-Nets in Base 16

(113, ∞, 256)-Net over F₁₆ — Constructive and digital

Digital (113, m, 256)-net over F₁₆ for arbitrarily large m, using

net from sequence [i] based on digital (113, 255)-sequence over F₁₆, using
- t-expansion [i] based on digital (75, 255)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 75 and N(F) ≥ 256, using
    - F₄ from the tower of function fields by Bezerra and GarcÃa over F₁₆ [i]

(113, ∞, 258)-Net in Base 16 — Constructive

(113, m, 258)-net in base 16 for arbitrarily large m, using

net from sequence [i] based on (113, 257)-sequence in base 16, using
- base expansion [i] based on digital (226, 257)-sequence over F₄, using
  - t-expansion [i] based on digital (225, 257)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 225 and N(F) ≥ 258, using
      - T₈ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(113, ∞, 340)-Net over F₁₆ — Digital

Digital (113, m, 340)-net over F₁₆ for arbitrarily large m, using

net from sequence [i] based on digital (113, 339)-sequence over F₁₆, using
- t-expansion [i] based on digital (101, 339)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 101 and N(F) ≥ 340, using
    - Drinfeld modules of rank 1 [i]

(113, ∞, 1741)-Net in Base 16 — Upper bound on s

There is no (113, m, 1742)-net in base 16 for arbitrarily large m, because

m-reduction [i] would yield (113, 5222, 1742)-net in base 16, but
- extracting embedded OOA [i] would yield OOA(16⁵²²², 1742, S₁₆, 3, 5109), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 565 116343 423307 542258 420872 475696 125346 645509 300721 874383 079647 180554 334069 226998 271377 321673 920395 618805 436499 335240 028360 744757 659290 755680 428950 027212 364568 413166 466077 596070 933561 131677 777989 766521 405920 156649 276864 708086 056962 879641 989623 723777 784964 798322 093542 386650 393103 229895 377825 608879 130420 605080 620117 126120 476994 809313 028439 528521 809342 496512 138893 155650 393577 850522 300113 116490 996467 363483 869615 980210 724466 527057 813777 417473 756274 973280 660183 424773 783135 245767 928374 999687 932597 128469 622976 786298 677359 331800 276881 099328 257451 684022 361762 425497 882844 862286 527456 418753 918617 800579 339995 895428 900137 228593 632592 043500 359579 790093 802488 429779 966759 209166 318981 826448 553753 604120 903049 270211 243117 311802 633744 425097 135086 609675 002017 868564 841744 589947 159808 308102 902099 097220 430820 565992 262888 658333 539191 042438 370339 383447 468945 739882 114817 320597 029045 129952 784996 769018 856168 679183 362869 518798 534640 983883 952121 975985 301583 158378 730040 373988 116955 348466 678850 403444 209925 637468 370642 732526 210129 830266 195101 836491 029530 861240 813741 774482 831520 183127 404735 559186 805912 771854 962030 244429 315556 399428 676895 950632 008712 827903 996073 232751 607789 774342 317089 901491 248475 557421 070253 256817 776265 765530 032417 108148 299018 396061 034267 480698 283959 976431 438905 625231 885911 148516 042816 369474 042770 574205 374943 870731 079097 625618 895094 004835 151213 694433 929666 277780 731799 186606 803562 852923 174138 582881 875673 990063 546720 545621 186356 607974 150448 949472 802675 640592 206936 232020 110411 596054 806111 294648 708786 409999 883884 812543 263152 285153 169476 735386 250469 222244 965578 371786 466487 161463 079773 517104 591871 636230 028835 362534 812078 511494 495183 881129 236658 961873 706272 649472 959213 937346 382969 597596 740116 054970 584608 186161 122332 798626 460424 238507 309547 577824 284961 520029 327362 943974 252438 895165 112237 936259 411517 398563 105998 393995 337644 622387 870325 167578 549930 278872 177575 259421 154762 477507 015655 898849 431545 691257 026108 515098 844876 206625 682660 647878 706403 319998 359391 992341 208147 042836 065325 555821 369835 065482 130914 900372 788716 133872 898580 759960 539817 539638 595412 836348 538852 341682 279975 911305 941448 000001 020301 024452 865379 203282 569966 486273 787855 103511 148658 535602 668451 364045 637883 261509 836962 729750 806413 063168 637384 516327 667538 775501 475323 935450 245942 767088 765070 027540 530894 405167 343996 712680 539126 182592 410107 702794 121644 698473 801322 297727 246116 499644 491799 349336 245485 238911 129205 737926 741236 629311 557155 268190 439466 600386 145098 648484 988686 813437 312552 730523 334821 442324 680826 763610 488683 905443 741557 360531 184225 393227 512478 731829 877002 227566 152938 201131 504877 655513 210738 592233 497488 023033 551754 646238 162259 644895 210242 317894 023508 169288 135292 178360 532674 553732 799710 595768 018311 523006 249172 217405 453771 492446 285094 444698 201280 095552 457063 808836 486242 806908 220300 301719 544596 892259 884811 761355 308171 488606 150778 571047 033862 422439 513114 704856 546745 836826 019078 031813 433337 431721 692915 624709 385493 760812 747745 498872 201341 529650 579102 867977 891620 814382 328637 977593 843135 391472 570956 773441 617469 038250 585651 918805 724803 592583 203585 783766 685927 639653 947678 847198 535924 379333 485537 178346 783775 277854 050144 420513 437775 861495 803360 549745 449518 246606 988337 246821 216164 697249 041244 690714 965229 604527 068374 916274 587588 438129 533289 391462 540779 062704 383535 897964 037199 199621 820935 790689 136452 834190 990571 753084 788740 736275 550965 665219 355060 260446 840023 993510 668586 828156 600195 187875 392231 845702 613424 959222 565002 414762 901630 431191 151489 061627 977737 858656 770652 750415 104218 078533 386560 219416 486485 370875 466167 388276 484283 283856 258740 839760 692151 830181 499828 624605 286281 208951 445392 966471 567391 101519 021943 058780 996254 760863 129159 442082 407883 056674 715842 548732 857556 679836 008383 766838 887450 254040 500030 010449 452304 239344 387190 496544 208503 695129 964300 183728 232678 388899 388216 657120 778309 231584 139988 627375 264096 293786 559835 050869 526084 336524 855965 674236 241582 785649 088179 174098 580614 401251 607150 452040 623526 963057 671685 065803 648626 120819 782407 273738 861508 964322 418333 741226 894180 834311 369105 219158 730299 705495 005255 213776 795307 144139 665497 232478 236177 253101 806233 402221 650671 684772 237505 005480 123816 683006 626025 262833 705234 645822 271208 252165 862087 581003 473448 227749 940438 784376 160159 402812 151238 532595 891247 458569 233192 734441 454624 952614 288244 338617 137939 938685 944024 358123 875195 573324 996847 243124 443459 998385 573415 015366 927897 593389 220731 735133 514173 131083 898510 333101 290740 679559 140730 591661 352280 109557 440682 835454 324273 173341 883888 480474 274779 679365 622583 638859 071448 724982 087440 759284 435290 296987 797830 019002 377484 399161 553666 146757 628088 693595 504799 794443 863501 739108 711539 062798 332520 808442 617749 569670 025022 168897 652082 422874 965194 504677 220424 152187 589295 794442 045553 159234 929961 059757 672130 550048 088899 276845 875858 477322 211145 743159 399310 172618 378445 494057 359398 445645 512036 185659 156063 542569 839079 730859 840420 860462 996915 505603 193791 984914 085598 870095 882988 292688 314925 031127 548887 454305 067729 917440 571744 438336 312651 321087 694899 507210 222405 056614 438912 159495 665424 075083 369234 974775 387206 437576 412351 782018 968853 472327 582533 762025 267313 368373 775334 237012 584497 965091 477557 261447 398421 889402 971717 281985 344874 778184 891053 143906 783640 884265 128843 439855 630242 706669 445162 647552 802968 217576 956229 798205 873466 911345 957271 083588 481061 636296 560947 379744 139089 511968 881932 116298 247389 649742 430340 055321 108446 212903 756782 998388 892365 650260 867546 750561 156821 157614 107856 897529 139879 341114 607626 410008 245943 101923 437050 491319 998712 018500 955990 143458 554548 259463 550610 981893 254961 986513 239051 956407 461529 093308 020236 920618 154132 376238 957550 229317 887942 615799 673420 210432 707509 171945 103609 641053 938597 108353 201597 574722 940549 963083 946908 803402 993730 260686 207996 465656 060201 423249 779387 998099 865162 234900 340514 348400 387009 955163 456831 723144 568325 047060 812398 665265 549541 304385 373535 540343 744603 106042 427231 619064 592716 708633 575335 326657 342390 714022 951581 234263 827231 071833 373574 627305 447414 704053 458864 616594 560081 465511 582500 825255 965284 322150 120524 021801 931940 830434 584213 657106 859197 758428 213778 599072 175360 636361 098775 920985 470426 017765 556542 561841 922173 872419 526906 745765 270517 778062 952518 186489 652706 283892 571175 637236 258414 033113 268970 070972 977951 447370 516606 170628 349850 608435 086517 681785 680088 507209 948018 905175 409825 733504 813912 094671 048547 650031 897645 871263 632698 186209 177075 471219 257283 961608 470528 / 511 > 16⁵²²² [i]

Best Known (113, ∞, s)-Nets in Base 16

(113, ∞, 256)-Net over F16 — Constructive and digital

(113, ∞, 258)-Net in Base 16 — Constructive

(113, ∞, 340)-Net over F16 — Digital

(113, ∞, 1741)-Net in Base 16 — Upper bound on s

(113, ∞, 256)-Net over F₁₆ — Constructive and digital

(113, ∞, 340)-Net over F₁₆ — Digital