Best Known (137, 137+∞, s)-Nets in Base 5
(137, 137+∞, 118)-Net over F5 — Constructive and digital
Digital (137, m, 118)-net over F5 for arbitrarily large m, using
- net from sequence [i] based on digital (137, 117)-sequence over F5, using
- base reduction for sequences [i] based on digital (10, 117)-sequence over F25, using
- s-reduction based on digital (10, 125)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- the Hermitian function field over F25 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- s-reduction based on digital (10, 125)-sequence over F25, using
- base reduction for sequences [i] based on digital (10, 117)-sequence over F25, using
(137, 137+∞, 210)-Net over F5 — Digital
Digital (137, m, 210)-net over F5 for arbitrarily large m, using
- net from sequence [i] based on digital (137, 209)-sequence over F5, using
- t-expansion [i] based on digital (127, 209)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 127 and N(F) ≥ 210, using
- t-expansion [i] based on digital (127, 209)-sequence over F5, using
(137, 137+∞, 567)-Net in Base 5 — Upper bound on s
There is no (137, m, 568)-net in base 5 for arbitrarily large m, because
- m-reduction [i] would yield (137, 2267, 568)-net in base 5, but
- extracting embedded OOA [i] would yield OOA(52267, 568, S5, 4, 2130), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 8869 864080 001872 157463 669515 529341 024207 133576 788796 848867 953290 739313 485447 887487 356979 659319 671735 059843 312471 082793 984628 603116 921932 420893 521669 645579 740340 260823 788339 392296 509552 397141 567109 197310 964103 473812 391624 961026 584908 763039 830591 842518 737586 117218 412895 157836 580509 091958 557156 147906 601376 809040 556483 668479 817012 770354 134936 822834 908010 343280 985956 066368 362435 140672 026232 566590 534040 526834 364654 959278 084287 182819 272747 114823 909522 824227 962713 464121 082358 042707 057788 837097 219178 684769 500873 918766 057794 467019 430776 696688 495935 961957 911997 818439 878604 180924 336212 318185 721675 388127 409267 242806 744018 469829 970369 811099 415387 956890 209170 826036 215875 920508 148702 457940 662064 580971 787278 063417 593487 163404 104999 947598 545595 140089 111878 127030 575059 201744 137536 962173 695935 016618 932812 939398 639404 503966 167041 345510 602804 477689 411242 610609 881595 973119 958117 944721 514300 440592 201305 508456 560930 804062 233544 244938 524405 355381 032377 605638 403686 328449 163225 906440 275096 962714 544262 411745 677938 750805 121791 526043 635747 139769 447790 033617 168730 216212 671247 717005 236326 737045 195176 549237 011972 827172 531516 452945 138804 052827 664328 470065 196840 691778 114295 753347 147850 468793 266794 801813 685626 830332 258540 079716 440446 917273 315724 039757 483618 583920 204548 106466 528817 893917 270496 885132 736921 313379 325856 444153 595951 781811 009927 061269 697562 353752 094670 250311 412147 790813 508866 518066 844213 896762 522649 446711 426055 660252 359761 347076 498192 903181 807962 392802 899558 191050 402239 990101 322008 996837 418724 975825 626669 094703 973008 069906 159314 323404 401003 158345 311028 210034 010044 182650 744915 008544 921875 / 2131 > 52267 [i]
- extracting embedded OOA [i] would yield OOA(52267, 568, S5, 4, 2130), but