Best Known (253, 253+∞, s)-Nets in Base 4
(253, 253+∞, 258)-Net over F4 — Constructive and digital
Digital (253, m, 258)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (253, 257)-sequence over F4, using
- t-expansion [i] based on digital (225, 257)-sequence over F4, using
- Niederreiter–Xing sequence construction 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]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 225 and N(F) ≥ 258, using
- t-expansion [i] based on digital (225, 257)-sequence over F4, using
(253, 253+∞, 301)-Net over F4 — Digital
Digital (253, m, 301)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (253, 300)-sequence over F4, using
- t-expansion [i] based on digital (234, 300)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 234 and N(F) ≥ 301, using
- t-expansion [i] based on digital (234, 300)-sequence over F4, using
(253, 253+∞, 777)-Net in Base 4 — Upper bound on s
There is no (253, m, 778)-net in base 4 for arbitrarily large m, because
- m-reduction [i] would yield (253, 3884, 778)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(43884, 778, S4, 5, 3631), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 6 961477 975125 923637 607988 230809 485621 456081 307007 648529 528961 671977 416670 149608 976775 229079 341992 536217 061870 929336 102759 272898 812311 145677 547075 261294 723452 346146 421521 018340 943127 341024 021822 977229 201499 399052 820808 070833 841155 142210 233391 144228 995385 947794 407189 610024 754515 595834 025929 753431 735889 888120 447297 450551 907563 786136 129111 738364 394536 119186 299924 416837 974341 954041 796873 073746 371430 890362 094271 096017 805459 219204 634309 548103 672568 259532 427449 267466 061858 120872 041440 970032 858852 025568 543803 076236 138702 697939 795286 296616 519459 554369 152571 703261 563760 078802 443514 893884 417659 304742 238226 446756 947375 564251 192927 350480 476012 065599 694413 967699 114310 785796 614722 467386 602661 873965 839848 596082 539691 119304 574702 503941 248267 554132 368125 931477 477989 712354 081643 447324 527114 414711 367291 290167 831933 296341 694175 397231 641388 458558 855060 391867 220061 295420 609707 730490 601661 103862 030152 443931 865796 781168 888919 969925 561818 686697 419706 445336 985349 611993 775033 930329 968485 509035 988906 573016 199730 049238 958795 763714 803548 083915 516102 040828 603907 805567 420064 903146 651177 906550 205557 785286 965750 095006 253443 235844 193584 785427 920995 777767 317976 401208 653079 185493 439643 960154 071687 412346 228156 231267 243299 843061 847377 764594 195374 505748 667655 238612 957876 067944 896632 554238 559077 189503 213614 418938 152843 111753 359036 838088 006856 236443 601126 181350 777606 792594 309028 971332 473327 798352 597291 540282 800709 139460 656534 018571 526197 971401 129034 381671 082950 836644 643173 560344 838726 857205 911782 292211 241861 538665 882045 140842 008525 027873 583823 023140 586348 242325 339712 226871 117121 614827 504870 861927 601842 063729 284316 272975 004003 397145 911655 353926 888023 565244 080916 975351 424590 121858 039944 384600 788226 473621 243189 542956 373489 383147 032197 294269 906538 074650 826232 423240 646286 892684 467272 267020 725058 575520 746668 601481 414820 657738 781214 394338 921668 219133 653668 348087 906757 909395 999257 358931 948250 860940 663792 757287 656883 722199 542150 717578 577833 988006 101110 736860 414356 681484 005501 529404 582385 122010 678936 155791 601994 532560 115475 927991 892709 056393 614715 729430 452911 036446 726948 258412 630188 390143 509408 214191 003835 516635 504152 810152 299337 922995 592721 605321 241658 408855 620821 230022 264799 984250 725558 326825 948369 568269 931764 831808 433715 341906 440218 542798 647420 500964 024567 940633 631933 295107 833342 732108 904801 420267 481895 664416 196668 490332 964579 363533 999584 601056 477184 / 227 > 43884 [i]
- extracting embedded OOA [i] would yield OOA(43884, 778, S4, 5, 3631), but