Best Known (172, ∞, s)-Nets in Base 4
(172, ∞, 200)-Net over F4 — Constructive and digital
Digital (172, m, 200)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (172, 199)-sequence over F4, using
- t-expansion [i] based on digital (161, 199)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 161 and N(F) ≥ 200, using
- F7 from the 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) = 161 and N(F) ≥ 200, using
- t-expansion [i] based on digital (161, 199)-sequence over F4, using
(172, ∞, 215)-Net over F4 — Digital
Digital (172, m, 215)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (172, 214)-sequence over F4, using
- t-expansion [i] based on digital (148, 214)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 148 and N(F) ≥ 215, using
- t-expansion [i] based on digital (148, 214)-sequence over F4, using
(172, ∞, 533)-Net in Base 4 — Upper bound on s
There is no (172, m, 534)-net in base 4 for arbitrarily large m, because
- m-reduction [i] would yield (172, 2664, 534)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(42664, 534, S4, 5, 2492), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 29040 543673 923252 331428 436287 070694 831332 843557 035725 678741 115728 923636 012146 410639 439112 065748 304977 332316 800768 759018 653454 684734 161690 344317 976659 055039 761190 251274 155388 709575 446550 341175 880719 892200 460685 450114 308962 001184 672366 322603 987104 348841 557451 666523 415127 225779 875005 954561 372207 067685 420921 938986 332290 852701 723637 342190 334256 673135 446514 409826 481952 291296 619153 993660 784432 607243 797376 524338 871220 676063 033920 244895 173537 911256 801485 898692 891400 708462 831157 441536 485223 880030 052345 880963 514333 358982 956911 294560 889785 693656 256746 168187 937404 287241 928847 756743 630119 034995 566871 777219 145777 690371 349474 501432 657948 541776 111809 142267 161260 139271 272663 900373 906955 642059 816051 392033 386667 301940 589827 303993 130471 451984 696962 796438 251323 557198 191993 387070 427660 803964 507254 727064 782199 364691 651095 595004 496445 830826 256157 920880 095203 651671 707009 740289 115559 786927 490297 955652 551236 237700 263056 294281 473712 889485 534414 971936 306708 503171 715809 047742 954565 705920 832453 384791 184834 195018 913010 461032 787195 902245 217666 940456 828229 601769 786718 876070 014853 175761 532182 923614 258673 628234 254017 235767 620631 744378 423589 873946 216806 787748 618064 416798 031427 818617 942971 718762 402481 949699 162296 279361 812191 762468 769856 731793 849630 044696 034134 608181 132864 375402 245438 447291 252636 205889 283453 248573 734578 957235 680846 040378 092571 561407 585223 246891 097239 211636 046367 225521 128453 024811 860722 642137 478844 501206 620847 348817 409481 006304 969019 505540 508388 662546 653953 429132 552484 544438 023865 898579 734337 202939 030885 219787 176875 201746 024846 895921 956121 361812 081519 291571 438009 035834 755252 818447 399799 654110 067955 859456 / 277 > 42664 [i]
- extracting embedded OOA [i] would yield OOA(42664, 534, S4, 5, 2492), but