Best Known (165, 165+∞, s)-Nets in Base 4
(165, 165+∞, 200)-Net over F4 — Constructive and digital
Digital (165, m, 200)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (165, 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
(165, 165+∞, 215)-Net over F4 — Digital
Digital (165, m, 215)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (165, 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
(165, 165+∞, 512)-Net in Base 4 — Upper bound on s
There is no (165, m, 513)-net in base 4 for arbitrarily large m, because
- m-reduction [i] would yield (165, 2559, 513)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(42559, 513, S4, 5, 2394), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 160 149958 607324 285401 652329 091134 629670 616918 367954 163311 491340 605721 950915 012351 963448 137462 542071 675843 284634 381039 273802 413205 406035 160209 566268 671189 655781 734436 871874 752836 451723 364464 905284 265176 155025 558216 813502 028710 845794 429962 789295 661733 443781 149693 138220 213003 427123 232447 212694 271473 530947 943393 071796 253381 664917 447697 222312 031976 852212 703217 714126 011163 705797 852836 043013 513700 490148 606646 312655 025559 696255 272632 189761 758094 244201 784971 890190 634412 630786 389735 170012 095722 782973 072741 131439 204676 771349 460761 849844 909797 548783 894899 890945 015843 499588 389889 415508 555796 353467 753091 775170 786403 953303 463156 892371 850028 322271 651061 819050 645825 965501 970547 480225 763987 490369 568559 490736 617230 430263 486815 900520 848077 868332 103356 242037 914283 513050 406420 574072 408037 453533 246442 853555 721464 988144 417835 858872 451317 460902 012988 295480 505816 988975 500555 693419 026314 684489 491379 858183 278128 794919 511103 991000 521330 176582 217537 305895 538125 716361 155519 168519 527613 403235 421177 679561 338655 000339 605257 670360 072478 963320 787671 396372 092747 170702 855237 043876 913415 721905 523869 230568 965694 223929 618176 687550 527976 249021 911488 880641 564680 848720 306031 544644 976619 490869 708334 175168 935485 367832 491776 864082 580605 432057 766096 413647 150219 129199 761386 960616 515386 158679 607454 877598 612988 929110 123604 310676 357416 254762 449302 208041 318878 579373 971480 371599 724975 743854 465444 029749 603954 959575 669998 140908 749989 304752 475389 895273 304155 755999 250165 895144 767098 624543 285912 184744 067863 534164 590193 918368 255179 129525 953564 096333 333818 889614 603613 462308 323328 / 2395 > 42559 [i]
- extracting embedded OOA [i] would yield OOA(42559, 513, S4, 5, 2394), but