Best Known (36, 36+∞, s)-Nets in Base 16
(36, 36+∞, 65)-Net over F16 — Constructive and digital
Digital (36, m, 65)-net over F16 for arbitrarily large m, using
- net from sequence [i] based on digital (36, 64)-sequence over F16, using
- t-expansion [i] based on digital (6, 64)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- the Hermitian function field over F16 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- t-expansion [i] based on digital (6, 64)-sequence over F16, using
(36, 36+∞, 66)-Net in Base 16 — Constructive
(36, m, 66)-net in base 16 for arbitrarily large m, using
- net from sequence [i] based on (36, 65)-sequence in base 16, using
- t-expansion [i] based on (25, 65)-sequence in base 16, using
- base expansion [i] based on digital (50, 65)-sequence over F4, using
- t-expansion [i] based on digital (49, 65)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 49 and N(F) ≥ 66, using
- T6 from the second 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) = 49 and N(F) ≥ 66, using
- t-expansion [i] based on digital (49, 65)-sequence over F4, using
- base expansion [i] based on digital (50, 65)-sequence over F4, using
- t-expansion [i] based on (25, 65)-sequence in base 16, using
(36, 36+∞, 193)-Net over F16 — Digital
Digital (36, m, 193)-net over F16 for arbitrarily large m, using
- net from sequence [i] based on digital (36, 192)-sequence over F16, using
- t-expansion [i] based on digital (33, 192)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 33 and N(F) ≥ 193, using
- t-expansion [i] based on digital (33, 192)-sequence over F16, using
(36, 36+∞, 576)-Net in Base 16 — Upper bound on s
There is no (36, m, 577)-net in base 16 for arbitrarily large m, because
- m-reduction [i] would yield (36, 1151, 577)-net in base 16, but
- extracting embedded OOA [i] would yield OOA(161151, 577, S16, 2, 1115), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 31 506672 433849 075877 749437 268541 732338 948686 500050 233343 672721 767124 319885 218264 336147 212905 211985 439196 359010 324811 638596 755138 578748 396552 480720 689665 254332 148173 749138 482431 963629 868178 087916 777666 721720 616456 889804 775547 154263 156635 870546 024027 930922 240061 693069 723976 534367 114631 711924 592994 083813 259758 146016 020329 081328 756018 803872 522949 443534 184931 696131 874137 537890 629848 670991 013819 648570 610317 939879 320112 009069 263534 250114 380184 993011 396696 002875 051185 678516 665692 895151 639627 695982 325142 350457 048589 392407 521313 375679 412124 529595 839214 842836 331795 029240 075110 663950 518415 128923 017112 014790 868109 681584 353159 930402 917240 841332 318631 170948 974700 878961 027318 461529 849011 378287 831274 208813 588241 860053 572506 225223 526537 062448 234214 405063 167328 566997 656335 263595 527353 495552 262054 403368 615591 524802 997152 709692 972511 942777 075756 424268 485696 700240 896149 417966 408739 268467 092755 135130 000337 835148 742207 026993 028526 016432 721798 816882 529526 981391 166730 961972 794369 622140 023897 193353 837288 400970 819570 092431 976145 322046 889412 113615 444175 064327 468328 272550 635497 005834 412650 277916 990510 571158 363035 847780 645844 301714 947761 862176 584201 000339 379307 371283 260479 515924 710008 599706 951897 034488 440850 907825 971499 432390 659179 015478 491878 643689 929643 164333 784096 608364 009672 165811 603507 749786 285178 118982 278650 314314 893137 293878 914394 264147 565690 444461 200184 140830 456939 208758 797590 936814 616576 / 31 > 161151 [i]
- extracting embedded OOA [i] would yield OOA(161151, 577, S16, 2, 1115), but