MinT - Best Known (31, ∞, s)-Nets in Base 16

Best Known (31, ∞, s)-Nets in Base 16

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

Digital (31, m, 65)-net over F₁₆ for arbitrarily large m, using

net from sequence [i] based on digital (31, 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]

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

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

net from sequence [i] based on (31, 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]

(31, ∞, 168)-Net over F₁₆ — Digital

Digital (31, m, 168)-net over F₁₆ for arbitrarily large m, using

net from sequence [i] based on digital (31, 167)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 31 and N(F) ≥ 168, using
  - a curve from the manYPoints database [i]

(31, ∞, 500)-Net in Base 16 — Upper bound on s

There is no (31, m, 501)-net in base 16 for arbitrarily large m, because

m-reduction [i] would yield (31, 999, 501)-net in base 16, but
- extracting embedded OOA [i] would yield OOA(16⁹⁹⁹, 501, S₁₆, 2, 968), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 303186 941489 116913 023894 652674 416013 532324 407051 452736 891344 793555 604093 465951 924674 834298 832958 806762 241637 399056 486929 884276 806967 533369 479814 436407 780232 886527 512220 898445 995566 324783 755860 390841 490717 257738 833243 140217 956564 144639 260213 553220 005760 553887 311343 792005 105141 127411 888479 159927 887545 883639 288948 550180 016668 466784 704845 849403 226058 162985 419512 302862 754595 140896 731743 148523 117393 929411 140370 610321 500873 823252 009992 237485 852174 526518 218766 301895 012390 387872 470175 268562 166592 879239 084391 870514 147197 103940 875405 797198 803845 366816 115046 007874 980957 443532 436860 380682 302104 957957 870489 656465 839340 538217 017431 748632 466896 128210 031511 243971 564339 203443 721350 206639 720102 723013 839343 972472 944956 784363 112595 311781 501331 147804 124780 269319 530480 268261 649307 613880 016381 763884 561944 210135 736041 976292 131134 359881 138319 563372 074843 449197 461589 775272 057099 988680 950716 122296 190004 186131 331568 431419 164014 507876 273740 880227 400090 662955 083327 461307 578084 818126 234534 215696 112337 655507 239717 175798 288451 941622 918993 655045 137244 825711 434998 283062 785459 116153 274469 669713 554759 600757 792848 678992 411702 093511 137476 687133 555912 417579 892718 854124 361084 213559 437027 225633 918277 774691 342617 605241 258951 835648 / 323 > 16⁹⁹⁹ [i]

Best Known (31, ∞, s)-Nets in Base 16

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

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

(31, ∞, 168)-Net over F16 — Digital

(31, ∞, 500)-Net in Base 16 — Upper bound on s

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

(31, ∞, 168)-Net over F₁₆ — Digital