MinT - Best Known (34, ∞, s)-Nets in Base 16

Best Known (34, ∞, s)-Nets in Base 16

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

Digital (34, m, 65)-net over F₁₆ for arbitrarily large m, using

net from sequence [i] based on digital (34, 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]

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

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

net from sequence [i] based on (34, 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]

(34, ∞, 193)-Net over F₁₆ — Digital

Digital (34, m, 193)-net over F₁₆ for arbitrarily large m, using

net from sequence [i] based on digital (34, 192)-sequence over F₁₆, using
- t-expansion [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]

(34, ∞, 546)-Net in Base 16 — Upper bound on s

There is no (34, m, 547)-net in base 16 for arbitrarily large m, because

m-reduction [i] would yield (34, 1091, 547)-net in base 16, but
- extracting embedded OOA [i] would yield OOA(16¹⁰⁹¹, 547, S₁₆, 2, 1057), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 328 904001 325280 969573 041866 905732 842505 822197 225339 440001 223008 352837 579476 855210 276812 067001 769954 012463 723844 452996 230068 685783 116315 806116 750344 200759 105093 438445 283558 784687 303750 933941 274032 799791 926019 645212 170839 724735 006062 382445 164155 841453 064756 207101 658946 100825 058546 828998 840478 246651 988641 928343 179492 498149 962052 229273 358760 389279 102079 981082 018103 555722 003413 036215 737367 375455 305621 853721 172454 116929 234796 424989 631726 246686 878067 029125 387423 326524 517011 067383 011519 525460 868683 955005 453493 035368 249454 304309 153031 953999 479976 362730 639491 136000 917781 740016 872077 470739 222756 248826 595171 730052 519377 021299 792664 807900 051809 752149 805790 541547 753272 616888 469337 863525 989631 946737 234030 221584 487899 121400 328107 069566 949904 096745 394634 198242 170567 744984 316915 336244 875205 165030 459342 382511 145160 098253 293134 009637 274493 620597 910846 918801 190062 291828 175299 381032 035555 896203 987235 993625 393893 976182 783389 988965 958861 360476 998621 500368 905294 718129 057322 021366 577716 294877 705420 471179 603616 496329 582398 869554 109974 967264 308192 534193 661309 422774 609716 033154 433030 482129 494374 191528 526616 687374 213255 095447 195078 660349 769488 375275 349127 972500 105210 964652 018082 236185 119969 816339 366581 299790 403102 372525 848533 910283 005931 173878 710035 724137 437761 221771 671530 603058 762340 183087 471063 139651 310384 994947 604649 345024 / 529 > 16¹⁰⁹¹ [i]

Best Known (34, ∞, s)-Nets in Base 16

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

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

(34, ∞, 193)-Net over F16 — Digital

(34, ∞, 546)-Net in Base 16 — Upper bound on s

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

(34, ∞, 193)-Net over F₁₆ — Digital