Best Known (69, ∞, s)-Nets in Base 9
(69, ∞, 165)-Net over F9 — Constructive and digital
Digital (69, m, 165)-net over F9 for arbitrarily large m, using
- net from sequence [i] based on digital (69, 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
(69, ∞, 192)-Net over F9 — Digital
Digital (69, m, 192)-net over F9 for arbitrarily large m, using
- net from sequence [i] based on digital (69, 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
(69, ∞, 578)-Net in Base 9 — Upper bound on s
There is no (69, m, 579)-net in base 9 for arbitrarily large m, because
- m-reduction [i] would yield (69, 1733, 579)-net in base 9, but
- extracting embedded OOA [i] would yield OOA(91733, 579, S9, 3, 1664), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 1 012662 938118 046545 137291 413862 873981 155516 902601 148048 013689 088380 406458 263736 818198 980124 532113 476997 375863 874017 666198 430018 365278 711912 486742 076304 327414 715548 887522 174032 762992 070775 101891 222161 234167 007336 310287 010802 555803 317500 011149 590599 513347 433512 672361 660138 940042 307156 173202 650888 566969 344619 087050 069974 859373 819445 946846 205917 417367 558689 655512 439818 060089 562593 118367 174330 257746 020687 349531 966056 554115 611553 294386 804230 256162 074649 170383 688342 643331 256574 888254 687193 271023 590398 587811 344614 863054 090858 533406 410050 908396 497200 854993 033432 464664 323689 283784 577824 206690 603827 783593 955935 892744 649734 757352 864027 113624 373694 619794 462021 090652 388587 070509 786729 624254 172974 993557 349973 408291 124878 793835 812581 375392 771195 881327 602193 187813 330055 557416 912716 862122 756262 532772 619981 214934 380637 526567 491038 447909 953640 707913 892199 335654 584085 204748 948117 269523 888411 111871 842145 511057 814932 040449 308939 518459 599793 602109 086675 765121 330574 373990 861355 912114 767125 554723 921173 277971 154550 385767 046451 793139 798908 352104 528488 875298 042706 106485 787496 182022 942925 246166 989580 115656 958017 663431 229701 070212 599592 494300 206868 373229 978820 728921 031177 328402 454284 388744 696783 927536 517178 926112 328476 631788 222804 214715 818320 823990 846728 527060 375013 861120 675191 509680 128747 757983 492239 097424 274141 703060 992504 252899 348491 950513 528516 216944 086184 929331 394101 114821 049585 498985 982521 885679 508226 322981 906197 011025 597866 539624 046479 073153 081811 823434 631523 673280 503748 531457 318202 206142 352330 384894 777185 190699 046644 328395 201164 484233 411975 496761 312899 829302 822848 397286 970929 580244 657838 479103 026726 152683 606950 803694 452001 515273 591908 170233 690871 601329 / 185 > 91733 [i]
- extracting embedded OOA [i] would yield OOA(91733, 579, S9, 3, 1664), but