Best Known (35, 35+∞, s)-Nets in Base 16
(35, 35+∞, 65)-Net over F16 — Constructive and digital
Digital (35, m, 65)-net over F16 for arbitrarily large m, using
- net from sequence [i] based on digital (35, 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
(35, 35+∞, 66)-Net in Base 16 — Constructive
(35, m, 66)-net in base 16 for arbitrarily large m, using
- net from sequence [i] based on (35, 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
(35, 35+∞, 193)-Net over F16 — Digital
Digital (35, m, 193)-net over F16 for arbitrarily large m, using
- net from sequence [i] based on digital (35, 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
(35, 35+∞, 561)-Net in Base 16 — Upper bound on s
There is no (35, m, 562)-net in base 16 for arbitrarily large m, because
- m-reduction [i] would yield (35, 1121, 562)-net in base 16, but
- extracting embedded OOA [i] would yield OOA(161121, 562, S16, 2, 1086), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 863 842152 577202 121892 036106 136933 892716 333402 572770 049826 208180 645592 351437 000811 031128 715866 087571 231231 983589 298082 772448 760553 337814 323917 969405 928175 960733 663151 201494 868530 335882 662395 191322 264467 773366 752791 982567 975080 364538 349340 763426 816376 164440 269428 475371 985459 813557 272365 367434 648970 315806 697177 582070 593568 377221 070772 348444 257764 240332 783446 520869 175550 822621 060417 895007 339704 644287 649116 973503 050979 702898 258161 172389 245495 685680 558987 613872 363611 000044 774867 258891 234914 636636 216995 094410 354154 254924 768389 079739 997995 011252 341542 781463 261155 851759 690015 716110 612854 792921 197455 853098 523987 911346 422747 435561 771557 806295 551611 707710 265732 555980 353263 762111 232104 210257 734668 925267 604889 297757 405859 214736 027030 805106 012647 560330 083531 233430 639806 279007 797411 566440 655412 943332 685728 706723 985144 904968 924788 350742 117091 252430 336196 695298 591046 608637 308817 756654 039210 693247 311565 700137 770085 445708 821470 828434 643297 739130 428725 152440 439872 786054 668208 459391 022577 913176 569240 255133 098941 286169 640373 171802 828870 007893 056400 887446 638250 990816 999872 506371 959820 331821 969937 231248 243891 096501 701448 678918 459574 568817 032985 496833 500210 346844 832936 705055 772855 570346 520478 720193 948599 984660 315902 804806 069028 241827 331452 131548 339118 797718 382437 208923 597067 248654 675344 761713 625775 621804 598887 358574 768723 202600 555473 326191 364960 659116 654592 / 1087 > 161121 [i]
- extracting embedded OOA [i] would yield OOA(161121, 562, S16, 2, 1086), but