MinT - Best Known (33, ∞, s)-Nets in Base 16

Best Known (33, ∞, s)-Nets in Base 16

(33, ∞, 65)-Net over F₁₆ — Constructive and digital

Digital (33, m, 65)-net over F₁₆ for arbitrarily large m, using

net from sequence [i] based on digital (33, 64)-sequence over F₁₆, using
- t-expansion [i] based on digital (6, 64)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ 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]

(33, ∞, 66)-Net in Base 16 — Constructive

(33, m, 66)-net in base 16 for arbitrarily large m, using

net from sequence [i] based on (33, 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
      - Niederreiterâ€“Xing sequence constructionÂ 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]

(33, ∞, 193)-Net over F₁₆ — Digital

Digital (33, m, 193)-net over F₁₆ for arbitrarily large m, using

net from sequence [i] based on digital (33, 192)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 33 and N(F) ≥ 193, using
  - a curve from the manYPoints database [i]

(33, ∞, 530)-Net in Base 16 — Upper bound on s

There is no (33, m, 531)-net in base 16 for arbitrarily large m, because

m-reduction [i] would yield (33, 1059, 531)-net in base 16, but
- extracting embedded OOA [i] would yield OOA(16¹⁰⁵⁹, 531, S₁₆, 2, 1026), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 1 560473 572282 399651 501508 652216 608231 914907 061553 465806 329065 690811 333208 707194 653116 047282 590270 790572 306424 436242 376907 478818 707521 725040 331867 008374 551741 500231 034344 165383 311083 871492 754046 235510 312927 270501 448693 138347 679583 576336 024351 772779 548964 743801 502110 436577 583478 381146 479389 162347 515826 216873 939233 539198 805455 097970 552863 219773 758717 571415 613579 461280 293488 505670 815652 934418 878692 395169 739167 965090 868555 343666 136453 498623 522726 207920 264144 273099 701930 811975 650215 555222 292494 554016 734895 310028 568080 680995 824150 857909 350239 112164 350328 322959 491522 439383 631183 528153 406245 509867 436714 387901 712885 744853 557458 255923 177147 337631 197786 432807 290363 690963 933129 596070 399339 392033 493295 506687 533196 213799 130224 451763 835721 963644 316443 537301 925068 064063 874137 658405 771433 022057 849493 830232 414845 154472 005035 906107 385352 478046 717408 011137 475284 144154 218173 934796 689800 163404 142872 702324 772685 301548 217987 464391 196643 029071 104926 043801 471577 561925 393328 842895 469678 862870 404314 867345 038796 289663 307598 628509 771986 271169 322016 779738 078355 392782 549308 364225 695867 657327 005010 429866 901701 386787 267123 447790 052406 088277 528148 567096 891654 593446 098795 766591 764150 305131 508158 024255 695048 762868 225808 939022 204146 876506 408720 877193 240780 340369 790239 710720 033719 750283 701614 804992 / 1027 > 16¹⁰⁵⁹ [i]

Best Known (33, ∞, s)-Nets in Base 16

(33, ∞, 65)-Net over F16 — Constructive and digital

(33, ∞, 66)-Net in Base 16 — Constructive

(33, ∞, 193)-Net over F16 — Digital

(33, ∞, 530)-Net in Base 16 — Upper bound on s

(33, ∞, 65)-Net over F₁₆ — Constructive and digital

(33, ∞, 193)-Net over F₁₆ — Digital