Best Known (251, 251+∞, s)-Nets in Base 4
(251, 251+∞, 258)-Net over F4 — Constructive and digital
Digital (251, m, 258)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (251, 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
(251, 251+∞, 301)-Net over F4 — Digital
Digital (251, m, 301)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (251, 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
(251, 251+∞, 771)-Net in Base 4 — Upper bound on s
There is no (251, m, 772)-net in base 4 for arbitrarily large m, because
- m-reduction [i] would yield (251, 3854, 772)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(43854, 772, S4, 5, 3603), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 24 196153 909961 376117 384728 273826 408734 443955 492488 340919 320201 657744 330487 622958 181061 024263 952339 736385 899402 636467 008664 545789 394419 161261 402500 836467 760353 276827 505812 708298 237127 161351 957572 553377 777518 008375 658480 629051 010362 326054 521325 369516 343134 143472 606472 723756 489630 058271 536763 021729 578430 263891 217263 825599 991511 509369 768203 133657 026701 525762 855176 665254 861108 027246 532770 812160 664530 740723 666156 420861 944992 189428 591201 129999 294302 246092 678394 501470 817925 858621 069444 675665 432400 613686 469296 737449 954557 704273 045643 094542 243586 273896 699792 230041 175946 422101 378471 589228 337897 015892 690615 572899 047910 599239 849884 596755 981405 437779 377872 772561 039225 750923 289618 457999 544872 599305 916660 717020 574587 958345 400249 524364 639796 446369 770855 994013 680887 386326 706625 440504 995434 400599 344302 753160 882597 077147 615384 968093 031584 534377 121762 149838 604026 673162 948327 424557 618972 065446 686013 774605 761517 277848 074336 457100 170202 000473 569006 525481 177331 076759 305487 144340 651798 684866 038722 844307 669064 809234 965625 591075 918190 518027 319679 612447 088522 072078 190252 541412 202969 318272 436617 736562 381490 091033 607421 372915 595307 045500 670593 892710 858332 242699 767176 704575 002978 980824 713945 351978 678976 672381 155045 248732 470206 747332 902609 210914 392317 867464 249143 081059 947913 852534 631251 649512 586035 799180 895117 529637 665204 675773 738724 150217 501865 164686 406201 765506 719227 321931 040502 291474 041604 090098 683826 768648 929070 701338 749588 027801 363473 037674 329573 389666 717685 931750 223412 353203 874058 703718 769714 840690 745240 493073 445913 844381 201878 037375 417770 028372 266764 234645 887695 658892 637647 764499 606774 981563 021900 216275 179559 407716 206434 277122 386886 093797 157211 401453 414721 541822 124212 172336 292831 339515 592919 083384 453499 631425 672157 181862 628591 200265 393530 241787 831290 464435 649239 624271 586576 031508 078862 466825 444698 117635 649831 142572 761789 321458 414428 116261 764256 596967 863215 860250 812997 049484 538161 936487 794317 463502 734560 368129 382567 934434 561531 743166 740821 887975 636794 324970 097836 376499 136069 428300 451248 943520 354984 245620 149316 824171 865303 390196 346313 170494 368794 778555 255742 540311 841302 185971 349254 512215 817618 335004 514939 994589 515036 243676 294080 824289 015663 396324 942198 384942 864603 552262 586441 403111 574044 929767 496114 390227 900203 474532 156994 135579 116441 753800 680853 539989 937172 398328 279665 399834 732578 627547 012538 494495 613830 299648 / 901 > 43854 [i]
- extracting embedded OOA [i] would yield OOA(43854, 772, S4, 5, 3603), but