Best Known (161, 161+∞, s)-Nets in Base 4
(161, 161+∞, 200)-Net over F4 — Constructive and digital
Digital (161, m, 200)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (161, 199)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 161 and N(F) ≥ 200, using
- F7 from the 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) = 161 and N(F) ≥ 200, using
(161, 161+∞, 215)-Net over F4 — Digital
Digital (161, m, 215)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (161, 214)-sequence over F4, using
- t-expansion [i] based on digital (148, 214)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 148 and N(F) ≥ 215, using
- t-expansion [i] based on digital (148, 214)-sequence over F4, using
(161, 161+∞, 500)-Net in Base 4 — Upper bound on s
There is no (161, m, 501)-net in base 4 for arbitrarily large m, because
- m-reduction [i] would yield (161, 2499, 501)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(42499, 501, S4, 5, 2338), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 121 048424 654348 811356 777468 463283 826495 424402 213315 173710 933068 957955 027155 520033 826228 238125 342177 810196 611949 625680 732928 922689 225972 292953 927306 167730 414380 411373 619048 646953 179445 695877 872958 453087 968225 044455 938016 311574 527338 809921 759313 315648 539614 568578 260693 438941 708406 483247 591907 686578 192447 810721 076114 258669 761821 581484 120696 563489 764072 297409 344577 348977 170989 114927 974570 169565 482973 631873 543759 665835 897983 408616 491881 511214 477505 603397 911514 492438 000624 545207 432983 185841 356578 940275 821963 165519 867574 229133 666086 254619 004636 184386 190328 060575 516104 445110 181380 713543 876148 909203 311167 927433 235909 623815 402659 277586 649721 545440 336870 000251 642103 013566 282544 887318 927437 107111 641884 590590 350532 748058 651755 597904 123966 733078 540890 796299 919102 900494 399758 753048 977542 705787 264454 878746 487771 109028 396823 599747 591016 494755 187865 004865 840525 334107 231257 144406 378097 164651 024584 484066 609866 505510 880964 239980 990928 969389 872278 873811 647284 153139 637329 187451 174595 473254 504209 719380 706573 230361 453638 614863 311281 829163 337517 165832 842761 551396 626756 375475 378601 103174 826892 810909 730948 063677 719182 880531 675063 018262 073169 550638 983308 386887 306578 903783 235183 289279 102487 201012 725865 954826 726621 123798 814609 104599 748966 928389 355414 299146 030939 647946 476667 738905 226477 311589 461828 227369 960831 142529 741082 432594 008689 127989 565081 331908 013394 194870 104087 212236 552703 693609 009978 323585 977318 259079 944676 724720 143389 800540 248190 182098 919615 354344 772321 367778 728784 278359 330264 674138 842642 910916 033466 335232 / 2339 > 42499 [i]
- extracting embedded OOA [i] would yield OOA(42499, 501, S4, 5, 2338), but