Best Known (88, ∞, s)-Nets in Base 7
(88, ∞, 69)-Net over F7 — Constructive and digital
Digital (88, m, 69)-net over F7 for arbitrarily large m, using
- net from sequence [i] based on digital (88, 68)-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) ≥ 69 from GarcÃa–Stichtenoth tower as constant field extension [i]
(88, ∞, 105)-Net over F7 — Digital
Digital (88, m, 105)-net over F7 for arbitrarily large m, using
- net from sequence [i] based on digital (88, 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
- t-expansion [i] based on digital (43, 104)-sequence over F7, using
(88, ∞, 551)-Net in Base 7 — Upper bound on s
There is no (88, m, 552)-net in base 7 for arbitrarily large m, because
- m-reduction [i] would yield (88, 1652, 552)-net in base 7, but
- extracting embedded OOA [i] would yield OOA(71652, 552, S7, 3, 1564), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 22 219478 673305 998119 904692 900224 499089 002022 854745 701409 305540 993191 540138 306179 176464 359463 861404 825879 604531 646033 114135 367471 211605 317156 123372 972286 762878 661426 747019 799675 314100 059025 743088 076541 359085 722579 577354 087670 015529 801428 654736 329093 514134 822344 269716 056176 747938 192580 602024 816349 172329 045189 375740 212142 333913 799122 266517 595585 346973 501973 315461 925312 870537 187273 103109 794226 862859 511569 068229 238468 214258 553904 715018 932623 046122 078359 919957 463989 739118 198729 861671 969432 007130 526067 453078 522392 049304 397142 607699 782697 162896 067187 151525 495016 212032 887322 798437 077533 510551 849994 826910 165353 972807 579735 509158 509985 269997 606782 448955 994765 324630 062819 298353 873598 728733 688403 658673 303603 259906 030040 727640 639685 794063 346476 042494 093559 880671 132994 165308 004739 371061 382829 687088 747674 178677 343078 190791 950055 881909 777825 235418 722107 776373 309015 514453 773218 625946 570538 264784 938882 910072 932319 810472 061729 198399 163013 884995 221330 788266 195644 771948 087585 987798 572863 344228 863427 620633 771018 792459 908804 427925 352263 428314 557350 068908 293255 240762 569451 850109 985501 878758 231740 936932 113005 734674 508626 596139 258714 643734 200111 888464 545469 938264 539687 578573 801869 853200 888439 090721 192192 613707 127221 202391 167835 892891 473907 054509 855765 823511 821094 503085 904555 038682 404965 610670 482014 158220 008496 725607 495938 278938 970513 167667 871992 702279 274681 054302 681335 694803 068272 205844 779281 321174 500157 / 1565 > 71652 [i]
- extracting embedded OOA [i] would yield OOA(71652, 552, S7, 3, 1564), but