Best Known (29, 29+∞, s)-Nets in Base 16
(29, 29+∞, 65)-Net over F16 — Constructive and digital
Digital (29, m, 65)-net over F16 for arbitrarily large m, using
- net from sequence [i] based on digital (29, 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
(29, 29+∞, 66)-Net in Base 16 — Constructive
(29, m, 66)-net in base 16 for arbitrarily large m, using
- net from sequence [i] based on (29, 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
(29, 29+∞, 161)-Net over F16 — Digital
Digital (29, m, 161)-net over F16 for arbitrarily large m, using
- net from sequence [i] based on digital (29, 160)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 29 and N(F) ≥ 161, using
(29, 29+∞, 470)-Net in Base 16 — Upper bound on s
There is no (29, m, 471)-net in base 16 for arbitrarily large m, because
- m-reduction [i] would yield (29, 939, 471)-net in base 16, but
- extracting embedded OOA [i] would yield OOA(16939, 471, S16, 2, 910), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 529714 724603 718140 650964 425025 175320 733728 843808 386648 413403 403295 802890 461991 912680 277496 556809 020660 806506 213791 005489 423094 434028 368523 541999 425392 456276 634189 021938 503984 932506 249847 142857 811147 393956 749606 006625 009462 684808 149998 486929 890604 620131 280565 731190 268096 903642 199022 661566 979560 868015 733425 071168 012348 939801 606451 673775 903418 632263 926661 461398 926712 208444 578175 326059 739707 712949 303983 365175 447279 647570 892804 485046 383549 750584 154088 788774 425324 079255 460428 387476 101868 079371 164050 471388 906344 855662 365544 140356 614973 144088 466097 426145 713042 241311 358600 478329 793622 340973 751669 759488 106445 021756 852674 292745 482593 908197 426910 101313 526727 325128 017146 181359 900069 918173 768124 762103 925305 702403 416887 060381 741598 626437 983006 769447 950667 353063 849533 094199 154304 915926 721077 174033 233975 585830 276040 057061 075413 260993 320938 896533 980812 250023 398349 937493 506986 100875 438667 466778 447500 425150 125106 825619 200386 300537 531280 196237 853761 433811 561400 469031 075128 706879 805752 053525 736822 142277 263174 638655 015691 685550 446300 790588 561238 449751 520730 787461 388552 545838 137334 765655 923532 702479 677255 260469 260804 086838 257221 437065 527296 / 911 > 16939 [i]
- extracting embedded OOA [i] would yield OOA(16939, 471, S16, 2, 910), but