Best Known (248, 248+∞, s)-Nets in Base 3
(248, 248+∞, 121)-Net over F3 — Constructive and digital
Digital (248, m, 121)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (248, 120)-sequence over F3, using
- base reduction for sequences [i] based on digital (64, 120)-sequence over F9, using
- s-reduction based on digital (64, 164)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 64 and N(F) ≥ 165, using
- T4 from the second tower of function fields by GarcÃa and Stichtenoth over F9 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 64 and N(F) ≥ 165, using
- s-reduction based on digital (64, 164)-sequence over F9, using
- base reduction for sequences [i] based on digital (64, 120)-sequence over F9, using
(248, 248+∞, 163)-Net over F3 — Digital
Digital (248, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (248, 162)-sequence over F3, using
- t-expansion [i] based on digital (151, 162)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 151 and N(F) ≥ 163, using
- t-expansion [i] based on digital (151, 162)-sequence over F3, using
(248, 248+∞, 511)-Net in Base 3 — Upper bound on s
There is no (248, m, 512)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (248, 3064, 512)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(33064, 512, S3, 6, 2816), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 3 753060 814646 139434 013571 990361 756131 641806 368033 013941 040015 155191 963518 566410 246947 767887 603234 392186 413007 183219 076973 645170 543427 774691 564493 479959 855555 864743 414054 558351 709016 196273 917436 182996 577407 059983 702160 126228 321734 589670 517963 141449 068377 122570 129503 433920 042209 552864 781653 990421 427702 195580 216299 432569 392964 531476 478309 270147 055829 813560 344142 882421 182359 215194 830305 358891 437723 480025 417431 625875 933183 690323 482670 120657 706016 220900 569531 100019 777269 974205 138044 485258 186447 778434 104590 919750 891830 433215 300791 718019 483436 906249 915518 577768 739569 587478 459354 568510 882747 048758 245894 559001 221128 373976 050954 414044 732568 721971 740519 106124 547697 657981 740679 777289 292243 877581 703851 686469 500598 272952 210983 249777 614608 959121 287386 382865 569624 182542 451131 645972 677797 336483 686882 904064 174440 432481 363911 286028 742701 896540 463584 813280 318724 527596 834611 578805 064063 962112 790947 769997 075395 570108 141155 023672 326329 538010 618719 629104 123891 212836 644128 370921 574874 947848 891952 655177 404261 360723 711106 414768 946713 237850 993242 433259 547287 295840 414214 525866 074842 150619 841600 289988 052914 548678 861767 184882 984166 979131 852002 141265 808299 296430 593391 779081 417287 389193 997944 457414 507469 730859 619785 163275 317074 148562 453519 610247 186044 570482 201051 335069 181152 945093 193367 267106 376092 531024 855421 709169 712970 427935 162751 710297 209485 943352 188096 243833 863959 549330 138264 158045 301293 940091 941346 850683 753313 883467 764899 209296 162655 850774 670603 201966 375007 326806 809913 / 313 > 33064 [i]
- extracting embedded OOA [i] would yield OOA(33064, 512, S3, 6, 2816), but