Best Known (32, ∞, s)-Nets in Base 16
(32, ∞, 65)-Net over F16 — Constructive and digital
Digital (32, m, 65)-net over F16 for arbitrarily large m, using
- net from sequence [i] based on digital (32, 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
(32, ∞, 66)-Net in Base 16 — Constructive
(32, m, 66)-net in base 16 for arbitrarily large m, using
- net from sequence [i] based on (32, 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
(32, ∞, 168)-Net over F16 — Digital
Digital (32, m, 168)-net over F16 for arbitrarily large m, using
- net from sequence [i] based on digital (32, 167)-sequence over F16, using
- t-expansion [i] based on digital (31, 167)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 31 and N(F) ≥ 168, using
- t-expansion [i] based on digital (31, 167)-sequence over F16, using
(32, ∞, 515)-Net in Base 16 — Upper bound on s
There is no (32, m, 516)-net in base 16 for arbitrarily large m, because
- m-reduction [i] would yield (32, 1029, 516)-net in base 16, but
- extracting embedded OOA [i] would yield OOA(161029, 516, S16, 2, 997), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 595745 886949 872724 754105 341973 462949 951963 714270 555565 225696 836878 567657 634820 004538 757190 194444 287855 762308 388780 726271 465477 846202 394054 336132 343965 829259 541978 121754 145706 839540 189136 792838 747506 731999 769012 204626 213547 417121 045763 326023 600392 235823 413070 225428 672691 336404 024294 924204 921931 604193 263253 861375 466194 801845 133762 009836 816382 491278 675979 750308 368452 443559 715520 757040 661689 046435 894060 852491 926999 996623 352155 815140 365008 916613 657668 964795 812768 376538 350361 301484 521991 558607 417572 658793 556862 693369 241885 832967 908287 287924 286105 276917 243093 343021 415451 077167 622854 358256 127115 751852 130355 122139 730753 082325 001161 711758 206248 437844 898400 375241 952510 945062 671032 639119 038932 680843 961166 947562 223298 483723 964629 117377 907503 281512 685910 028898 620913 343410 413531 661021 005172 289923 482310 045338 464192 803853 691150 746367 877440 502384 125284 611665 321474 596649 600436 086669 452236 041458 499633 513841 380361 019296 250728 009568 709261 555146 281354 615549 664026 820953 436349 577702 318628 393736 599916 942454 439695 772117 984060 790148 832529 755421 369829 395416 544328 356580 750299 003645 668627 990523 484887 000399 822574 405130 141603 433265 397592 032448 874681 878345 249527 536645 026593 068561 650051 932550 835492 756710 324425 358327 666640 665279 337344 623894 138327 707454 275584 / 499 > 161029 [i]
- extracting embedded OOA [i] would yield OOA(161029, 516, S16, 2, 997), but