Best Known (138, 138+∞, s)-Nets in Base 4
(138, 138+∞, 130)-Net over F4 — Constructive and digital
Digital (138, m, 130)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (138, 129)-sequence over F4, using
- t-expansion [i] based on digital (105, 129)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 105 and N(F) ≥ 130, using
- T7 from the second 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) = 105 and N(F) ≥ 130, using
- t-expansion [i] based on digital (105, 129)-sequence over F4, using
(138, 138+∞, 176)-Net over F4 — Digital
Digital (138, m, 176)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (138, 175)-sequence over F4, using
- t-expansion [i] based on digital (125, 175)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 125 and N(F) ≥ 176, using
- t-expansion [i] based on digital (125, 175)-sequence over F4, using
(138, 138+∞, 430)-Net in Base 4 — Upper bound on s
There is no (138, m, 431)-net in base 4 for arbitrarily large m, because
- m-reduction [i] would yield (138, 2149, 431)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(42149, 431, S4, 5, 2011), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 3 618346 296428 617804 521138 871658 709264 020113 135059 861615 150979 269357 279404 501890 777371 071124 027374 717287 812413 882901 100110 399328 062749 046812 296167 654024 812433 827474 329477 766215 531267 388991 432350 557767 845238 582831 600762 775671 775000 875456 450501 223827 258418 753973 613596 451549 326893 622180 159869 764904 477421 067833 661012 306644 753569 735336 773067 085937 295304 228568 979877 288011 522039 008536 293838 674056 774074 398719 596963 915168 671656 220209 467908 142499 524225 343518 622171 792691 533765 450547 138836 119122 987721 099785 736322 994091 810448 633068 388239 372707 454849 384319 033466 156336 249304 466110 034325 489163 181238 881327 594301 098371 510920 532564 894038 105114 978283 286785 159052 666047 410221 905846 989096 438072 652870 066635 594626 319291 774737 494592 156839 878770 250574 170680 606824 205480 244092 156645 864468 852775 476538 127780 154897 535460 945545 163655 984912 762333 944504 239977 064066 006802 916625 161057 580709 524405 235839 449225 263349 664039 669710 341014 921293 258692 056273 996848 363079 719086 076403 519690 970546 164225 629729 515065 043635 759090 296992 201934 822028 037390 913480 014516 907317 797165 837692 066088 477792 658092 357343 669328 147480 493095 156515 210274 027928 503205 668004 158109 503547 868870 043416 588279 602119 406743 845058 803353 463317 296614 741646 030694 935583 014533 675096 790620 155667 113928 621485 534663 234162 181283 236801 822200 716943 863895 507813 866370 236416 / 503 > 42149 [i]
- extracting embedded OOA [i] would yield OOA(42149, 431, S4, 5, 2011), but