Best Known (104−89, 104, s)-Nets in Base 25
(104−89, 104, 126)-Net over F25 — Constructive and digital
Digital (15, 104, 126)-net over F25, using
- t-expansion [i] based on digital (10, 104, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- the Hermitian function field over F25 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
(104−89, 104, 140)-Net over F25 — Digital
Digital (15, 104, 140)-net over F25, using
- net from sequence [i] based on digital (15, 139)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 15 and N(F) ≥ 140, using
(104−89, 104, 1321)-Net over F25 — Upper bound on s (digital)
There is no digital (15, 104, 1322)-net over F25, because
- 1 times m-reduction [i] would yield digital (15, 103, 1322)-net over F25, but
- extracting embedded orthogonal array [i] would yield linear OA(25103, 1322, F25, 88) (dual of [1322, 1219, 89]-code), but
- the Johnson bound shows that N ≤ 1 211626 926466 429827 148531 667575 965986 787019 995320 743358 795843 398235 139157 375034 671125 280958 510398 078128 040606 181588 084921 044170 652117 324347 885263 990831 230335 909531 219682 984360 687480 000790 658043 542455 676171 586146 292679 320912 243102 527002 460861 522415 747455 504616 189787 567293 678993 971007 171402 376529 523132 851027 993076 189333 985726 657863 956277 911810 479775 357865 358837 719521 873788 060002 803951 497409 690877 547887 357899 492086 336947 141199 520556 448345 557056 711314 918819 104915 013268 014575 588718 950816 565831 158416 058326 865488 504734 303911 258311 404817 211322 577009 110873 778060 133619 505184 518175 258610 813293 537241 212600 385087 759265 529795 786677 954475 096911 046712 246357 475230 902917 991530 749159 674138 109209 476836 469768 295433 731356 489115 842642 084321 115407 254236 715858 517272 309350 205451 075333 793388 951217 178383 876154 736890 501049 441778 994586 658665 480383 393276 676840 982874 390694 566470 646063 863612 747649 375292 733390 777548 469662 480853 865545 844999 939774 037836 664706 041432 769994 117933 717267 308257 300064 707966 106204 764468 456615 694662 350774 776035 236317 216419 342475 820227 682268 957059 884456 475153 270350 530478 403469 064660 207501 956606 204375 074549 527322 894287 783838 130714 568030 829353 096507 524385 978272 238646 953794 815387 971251 810324 585364 635529 884068 314069 119410 091601 937565 872360 382740 306123 845386 288434 917281 822431 204921 041265 692252 592406 274111 511523 805235 355266 883890 178463 286117 212492 910259 090696 624318 577236 949001 147971 281789 552458 660598 240817 577542 725153 758836 183130 553643 500641 339435 455008 253964 957931 967184 276827 632699 095838 245768 847081 214121 337883 615179 606849 938615 306308 293584 065449 523852 745057 275362 072490 129302 720668 413577 798567 684833 698743 006985 114866 000266 920980 265150 207227 029032 705231 294972 724970 721369 760837 173905 232970 999044 < 251219 [i]
- extracting embedded orthogonal array [i] would yield linear OA(25103, 1322, F25, 88) (dual of [1322, 1219, 89]-code), but
(104−89, 104, 1323)-Net in Base 25 — Upper bound on s
There is no (15, 104, 1324)-net in base 25, because
- 1 times m-reduction [i] would yield (15, 103, 1324)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 1 000740 394299 906545 474938 429722 277523 608411 700041 227146 351057 875102 329336 920732 791819 654398 132404 185117 723259 209146 961513 694262 360109 443280 761985 > 25103 [i]