Best Known (25, ∞, s)-Nets in Base 16
(25, ∞, 65)-Net over F16 — Constructive and digital
Digital (25, m, 65)-net over F16 for arbitrarily large m, using
- net from sequence [i] based on digital (25, 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
(25, ∞, 66)-Net in Base 16 — Constructive
(25, m, 66)-net in base 16 for arbitrarily large m, using
- net from sequence [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
(25, ∞, 144)-Net over F16 — Digital
Digital (25, m, 144)-net over F16 for arbitrarily large m, using
- net from sequence [i] based on digital (25, 143)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 25 and N(F) ≥ 144, using
(25, ∞, 409)-Net in Base 16 — Upper bound on s
There is no (25, m, 410)-net in base 16 for arbitrarily large m, because
- m-reduction [i] would yield (25, 817, 410)-net in base 16, but
- extracting embedded OOA [i] would yield OOA(16817, 410, S16, 2, 792), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 541 469259 974676 085558 625549 174807 926720 083912 734004 666911 121844 777982 505931 701665 350559 692883 115529 729674 231329 218514 255111 331835 882737 207752 475480 068054 412155 857147 602704 062771 589695 511480 392283 461132 561207 645710 150613 432904 958655 596888 167139 115195 299227 351116 945234 795082 650782 767789 292739 487211 183582 837928 840301 373907 083219 662990 379847 421811 239448 756700 496905 862110 628522 492603 343422 169392 057649 735464 097501 138019 742115 319642 285525 236121 352117 644781 550459 986642 642578 708719 695086 424647 929619 793746 181369 621396 454250 332842 194933 948800 184327 869316 096541 331867 040224 800418 674752 669677 430803 744697 833841 528363 453695 754055 912323 268385 273962 705875 251354 085344 221070 579865 707540 712691 768197 315359 432137 660999 789789 243097 105466 301940 021010 441719 429148 020662 934240 078867 314705 329650 487592 553613 412468 907347 615874 360190 605827 058680 472829 089539 235016 769296 054468 753559 933369 416658 681323 583672 249189 510253 454067 895856 279260 361050 064981 182491 748803 246518 104458 836252 880897 317729 386571 271784 890368 / 793 > 16817 [i]
- extracting embedded OOA [i] would yield OOA(16817, 410, S16, 2, 792), but