Best Known (67, 67+∞, s)-Nets in Base 9
(67, 67+∞, 165)-Net over F9 — Constructive and digital
Digital (67, m, 165)-net over F9 for arbitrarily large m, using
- net from sequence [i] based on digital (67, 164)-sequence over F9, using
- t-expansion [i] 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
- t-expansion [i] based on digital (64, 164)-sequence over F9, using
(67, 67+∞, 192)-Net over F9 — Digital
Digital (67, m, 192)-net over F9 for arbitrarily large m, using
- net from sequence [i] based on digital (67, 191)-sequence over F9, using
- t-expansion [i] based on digital (61, 191)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 61 and N(F) ≥ 192, using
- t-expansion [i] based on digital (61, 191)-sequence over F9, using
(67, 67+∞, 562)-Net in Base 9 — Upper bound on s
There is no (67, m, 563)-net in base 9 for arbitrarily large m, because
- m-reduction [i] would yield (67, 1685, 563)-net in base 9, but
- extracting embedded OOA [i] would yield OOA(91685, 563, S9, 3, 1618), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 1446 661596 189705 300978 837180 868259 429566 391154 657278 312297 200859 831841 957599 452992 772821 208747 052620 899413 251734 095231 610338 539308 728711 140178 385477 956732 587936 562910 096230 305484 895005 626633 855988 961506 015721 473745 894872 219952 522614 498077 944277 328507 083133 091118 113289 930523 213352 295084 857660 231791 298800 417039 041348 295002 056619 979132 502716 008973 645539 915107 577695 057475 215149 573901 256293 070958 187346 405811 905513 512321 595042 900166 126973 980360 352727 914100 504749 060449 814617 828524 847629 237806 336820 365749 580626 664047 347160 402063 764820 767545 847010 006832 354227 924719 861377 759891 932327 532806 425420 455426 350545 088202 792342 439908 005404 095650 870698 470300 563928 918359 877959 355714 893300 223320 552622 634202 617204 845675 800403 624906 277080 046554 188598 320344 114041 975040 010726 396242 935457 423381 979512 099175 578233 118417 713581 476899 642528 494579 379819 603058 367525 178897 590428 175515 913618 151363 866294 521997 149602 604396 992545 295054 369290 024858 933115 852300 022965 197938 385002 557032 437496 638981 978319 583661 755297 878985 651990 018470 844909 990193 952871 824695 691149 538461 836371 913923 808092 702327 163610 948117 748110 359079 840563 093132 648808 188493 483608 156910 573462 716028 974799 612115 153102 804678 871355 707198 180344 828512 277262 225814 766078 085310 159302 056485 240035 842917 828210 518107 875065 878964 075717 365896 970232 714124 672015 410741 185214 407990 979601 142727 563205 658080 675612 262015 458437 348495 455625 263607 119594 607602 097621 654334 173993 229315 898007 514191 100691 612736 970481 282588 181569 788529 112128 448545 496285 206408 970188 880638 970702 572596 619665 143481 029051 831220 952047 564797 877129 522836 436392 058388 596588 872257 885909 787122 379453 628674 947382 869102 644123 / 1619 > 91685 [i]
- extracting embedded OOA [i] would yield OOA(91685, 563, S9, 3, 1618), but