Best Known (228, 228+∞, s)-Nets in Base 3
(228, 228+∞, 112)-Net over F3 — Constructive and digital
Digital (228, m, 112)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (228, 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
(228, 228+∞, 163)-Net over F3 — Digital
Digital (228, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (228, 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
(228, 228+∞, 471)-Net in Base 3 — Upper bound on s
There is no (228, m, 472)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (228, 2353, 472)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(32353, 472, S3, 5, 2125), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 5217 529074 234589 790020 078901 707615 701586 498686 337371 533833 875488 793242 055000 356016 383601 142673 259376 972442 877942 628457 397540 066579 500577 560314 419230 842407 929579 192996 435729 561325 961244 499951 026890 015543 898863 168361 609596 268172 940550 131928 719630 930222 425663 965697 008756 627218 525930 045411 699094 803637 040367 708113 593989 233653 954823 451034 560305 378818 452409 823611 363065 840888 248135 740551 031193 719623 138383 870127 858425 391610 557223 224572 638464 923368 453554 380216 356559 535401 557696 486883 209899 528144 647431 796061 432914 577312 678563 506415 595036 956451 879174 348350 996097 019809 300981 153726 353931 430508 269156 755841 996961 049161 325997 767071 826906 672235 211264 294558 968433 071266 107134 212968 259578 837372 434981 599890 915132 317206 641664 621197 687780 472080 002184 397959 413049 686272 828480 567209 573784 740025 778067 659937 467380 526560 892081 737772 014878 032135 255654 924190 130455 903381 698434 094929 138198 533005 031265 529636 755364 122989 251599 777123 196470 628890 005184 072271 012419 709695 512702 452522 782238 312677 635346 356118 272791 237449 896455 348224 796853 651442 034436 880229 657276 933150 922311 537768 972977 211196 317987 258693 655084 274486 622180 402090 535112 929881 383651 563375 / 1063 > 32353 [i]
- extracting embedded OOA [i] would yield OOA(32353, 472, S3, 5, 2125), but