Best Known (139, 139+∞, s)-Nets in Base 4
(139, 139+∞, 130)-Net over F4 — Constructive and digital
Digital (139, m, 130)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (139, 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
(139, 139+∞, 176)-Net over F4 — Digital
Digital (139, m, 176)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (139, 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
(139, 139+∞, 433)-Net in Base 4 — Upper bound on s
There is no (139, m, 434)-net in base 4 for arbitrarily large m, because
- m-reduction [i] would yield (139, 2164, 434)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(42164, 434, S4, 5, 2025), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 7755 923290 087973 717132 715647 693220 490394 967612 152643 332073 512436 524821 530840 488979 338913 567525 875148 501631 979324 038599 289613 971891 536752 550764 358829 509509 601663 138680 005524 716423 171259 015248 288474 032371 692333 198109 874842 475397 435768 446912 150788 071034 660200 627532 800288 298842 352448 281729 657747 897550 305242 802069 322174 697118 885651 186852 446834 462440 819142 456364 814222 956615 116376 619675 337652 540937 276141 100449 795976 158961 911807 643409 783903 580101 635393 932285 819683 298571 599457 449954 206713 068269 337287 601208 610038 902981 798600 039654 741475 693287 172341 571168 250927 078862 245628 722508 087005 933010 589641 969680 622484 725041 581489 138185 635951 172071 819108 745378 613019 619849 008216 106677 022177 477112 932937 332675 493710 972241 365569 557477 069699 007824 704655 012054 048062 675187 202459 723751 537741 782694 881628 824401 565072 561446 784900 332895 706216 323630 802131 376193 286674 072417 576340 455451 347972 944672 132190 744607 921708 182817 379476 225627 818717 382110 755392 158128 438040 534113 225599 269148 108519 238163 980575 715493 812503 012226 936298 468157 357414 222729 821479 230102 529613 196147 026175 294203 456304 416757 404238 957331 229952 375888 927894 476799 698792 392654 433167 586005 600734 980357 259935 583910 897614 478139 304959 332848 609420 270610 781151 002887 513217 322812 848326 602312 563999 686215 976808 035090 745719 212876 301247 951840 333624 204550 739670 928875 735960 518656 / 1013 > 42164 [i]
- extracting embedded OOA [i] would yield OOA(42164, 434, S4, 5, 2025), but