Best Known (26, ∞, s)-Nets in Base 16
(26, ∞, 65)-Net over F16 — Constructive and digital
Digital (26, m, 65)-net over F16 for arbitrarily large m, using
- net from sequence [i] based on digital (26, 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
(26, ∞, 66)-Net in Base 16 — Constructive
(26, m, 66)-net in base 16 for arbitrarily large m, using
- net from sequence [i] based on (26, 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
(26, ∞, 150)-Net over F16 — Digital
Digital (26, m, 150)-net over F16 for arbitrarily large m, using
- net from sequence [i] based on digital (26, 149)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 26 and N(F) ≥ 150, using
(26, ∞, 424)-Net in Base 16 — Upper bound on s
There is no (26, m, 425)-net in base 16 for arbitrarily large m, because
- m-reduction [i] would yield (26, 847, 425)-net in base 16, but
- extracting embedded OOA [i] would yield OOA(16847, 425, S16, 2, 821), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 117 887809 354226 939273 083539 796488 005055 147222 383512 541410 978825 641036 633184 946853 275051 170923 040649 646152 671396 881133 284270 517161 742700 861403 677304 075020 188848 720043 462257 983691 928543 217530 635436 422232 683433 801205 494066 325620 668814 637690 272054 078906 815553 538706 565128 810415 596956 736816 695169 481617 309567 961794 341399 099506 673141 506640 342190 019804 929634 985156 909214 012168 022274 411955 976641 087037 385179 159474 002305 167163 991037 315793 887104 986564 707315 848083 141671 798857 742605 453781 396611 799838 077868 268920 630208 551797 299992 151089 665204 289238 252402 862059 165112 485523 061448 533831 459644 167240 282558 271045 929733 101974 824585 223242 559005 812253 019655 471916 921471 588904 504660 197264 777403 178920 000836 482947 930823 565350 331546 282230 764700 160261 708981 117005 474150 683309 738739 274632 511240 269144 377235 331076 390544 195443 616180 117759 505663 643587 756738 639064 261104 610601 680078 310330 091452 127905 159356 319856 057250 310082 120058 018377 598097 200991 902731 199356 593289 672887 149338 094839 029686 440802 793756 059943 850339 211941 655604 441800 249375 558060 736512 / 137 > 16847 [i]
- extracting embedded OOA [i] would yield OOA(16847, 425, S16, 2, 821), but