Best Known (81, s)-Sequences in Base 7
(81, 61)-Sequence over F7 — Constructive and digital
Digital (81, 61)-sequence over F7, using
- Niederreiter–Xing sequence construction III based on function field F2/F7 with g(F2) = 21, N(F2) ≥ 2, and N(F2) + N2(F2) ≥ 62 from GarcÃa–Stichtenoth tower as constant field extension [i]
(81, 104)-Sequence over F7 — Digital
Digital (81, 104)-sequence over F7, using
- t-expansion [i] based on digital (43, 104)-sequence over F7, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F7 with g(F) = 43 and N(F) ≥ 105, using
(81, 508)-Sequence in Base 7 — Upper bound on s
There is no (81, 509)-sequence in base 7, because
- net from sequence [i] would yield (81, m, 510)-net in base 7 for arbitrarily large m, but
- m-reduction [i] would yield (81, 1526, 510)-net in base 7, but
- extracting embedded OOA [i] would yield OOA(71526, 510, S7, 3, 1445), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 125 364784 178677 917987 055463 253769 107871 196624 684315 329680 921441 728855 829543 888000 460279 201355 053375 213060 070960 833700 372455 253824 772596 163877 202963 558689 111367 477617 368062 472558 788726 901096 816668 532841 375539 637669 277870 237329 461895 518805 355724 663804 742762 936624 950714 060141 152927 364063 697591 494332 685012 818020 270680 032824 017787 016255 523402 942958 192161 465132 031714 609257 600667 106181 186271 705378 168344 947135 412092 116612 430725 468267 506864 998722 331588 495372 010408 927959 562165 839168 430157 654308 110223 952282 775523 004054 289156 858487 143570 626805 956544 279448 032669 039770 314133 990704 620059 782108 127018 165772 654654 792917 196873 832922 300490 284902 912735 918382 080083 535407 998522 439994 225260 329804 108160 075951 351747 565466 859271 661808 429146 655313 389475 789813 592707 957826 961299 701464 593367 834019 252164 614194 853411 933043 418256 161761 637783 930832 308797 041044 847003 466931 464452 815208 518739 193717 075969 379128 092094 942169 912453 397374 004158 294245 104481 895933 595601 224499 802443 379950 792839 679375 151420 984187 631829 138224 403979 802431 038643 163304 180214 667548 484330 870367 328518 485095 945932 699072 547738 761077 303623 650572 492793 692780 450388 386555 311369 517142 239387 471190 247272 127481 828266 268501 605446 663383 973228 380959 890535 353738 996804 304094 763105 807258 135633 253066 570307 213974 201047 342659 916852 288883 815298 040511 814440 100349 / 241 > 71526 [i]
- extracting embedded OOA [i] would yield OOA(71526, 510, S7, 3, 1445), but
- m-reduction [i] would yield (81, 1526, 510)-net in base 7, but