Best Known (226, 226+∞, s)-Nets in Base 3
(226, 226+∞, 112)-Net over F3 — Constructive and digital
Digital (226, m, 112)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (226, 111)-sequence over F3, using
- t-expansion [i] based on digital (189, 111)-sequence over F3, using
- Niederreiter–Xing sequence construction III based on function field F4/F3 with g(F4) = 79, N(F4) ≥ 2, and N(F4) + N2(F4) ≥ 112 from GarcÃa–Stichtenoth tower as constant field extension [i]
- t-expansion [i] based on digital (189, 111)-sequence over F3, using
(226, 226+∞, 163)-Net over F3 — Digital
Digital (226, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (226, 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
(226, 226+∞, 466)-Net in Base 3 — Upper bound on s
There is no (226, m, 467)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (226, 2795, 467)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(32795, 467, S3, 6, 2569), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 9236 908624 600192 057871 203714 094278 951263 485547 766049 622970 520359 224059 742406 801632 699408 003579 715643 502875 767324 236446 217642 898071 935067 299404 190070 103390 405727 200836 009873 280663 205695 455351 638371 270950 053251 491132 647868 984617 646351 812141 409142 269721 648821 490614 304671 358395 882016 210626 023389 792304 233175 078428 758926 826272 233121 320583 361324 701592 751222 013063 701637 541967 540107 838796 078254 656831 755045 283608 918630 251203 318926 743537 823240 228896 016834 912885 789660 504352 146211 647347 489137 738654 809954 878737 075359 831614 532627 340878 300711 499534 536627 376369 241989 336696 167939 274973 998192 456162 403186 065992 692297 278849 617849 455002 766251 918501 447853 475013 208994 142571 897836 388871 565443 837877 931936 898217 685337 087756 217184 638256 600610 468676 054926 436475 464429 023994 349161 442738 661323 130094 104119 293569 758381 059396 345269 706204 628447 752800 082402 487724 952931 735639 767263 079474 232834 413215 168568 752753 918732 927528 796101 403316 337094 245295 445944 912490 652085 777103 573874 231230 101757 366388 952045 192987 131991 233360 908761 514886 999313 862654 618836 708242 204893 878192 689667 134685 915119 885337 499094 212070 503802 461937 271912 994488 378476 087921 062896 331361 118292 475443 066797 407427 412965 260633 784273 696623 902159 102178 365590 988608 235771 830141 824435 301318 556411 290981 206651 007505 974646 634817 222522 303462 493183 474878 769885 972800 476383 052821 241282 513660 681406 039219 567606 / 257 > 32795 [i]
- extracting embedded OOA [i] would yield OOA(32795, 467, S3, 6, 2569), but