Best Known (106−91, 106, s)-Nets in Base 25
(106−91, 106, 126)-Net over F25 — Constructive and digital
Digital (15, 106, 126)-net over F25, using
- t-expansion [i] based on digital (10, 106, 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
(106−91, 106, 140)-Net over F25 — Digital
Digital (15, 106, 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
(106−91, 106, 1317)-Net over F25 — Upper bound on s (digital)
There is no digital (15, 106, 1318)-net over F25, because
- 1 times m-reduction [i] would yield digital (15, 105, 1318)-net over F25, but
- extracting embedded orthogonal array [i] would yield linear OA(25105, 1318, F25, 90) (dual of [1318, 1213, 91]-code), but
- the Johnson bound shows that N ≤ 4872 852408 431482 401658 366878 969790 037184 784632 107568 293981 530721 470598 015240 577737 415990 406522 660881 256467 475576 589096 855112 180568 001329 026015 842962 252671 914511 009361 316501 396864 938566 775000 678859 765964 439094 861367 474816 480010 251036 006591 273838 657733 072234 733061 577191 954985 940602 710467 958058 703022 275094 501542 477709 991063 311624 127776 361941 444231 842603 415664 605685 228934 283469 028513 312780 443042 516199 560737 840521 159682 975274 288441 462836 822009 146555 441384 613776 347268 555806 987429 876272 835583 223075 835504 675425 892866 465345 436354 103591 470574 299579 604579 015332 276173 498592 521335 618053 118441 600233 606754 088750 632870 841972 651506 745887 775009 365598 790894 532209 507810 887411 306861 845639 068690 094180 985826 626531 022011 527703 996977 696659 666319 532229 668289 834792 111971 979226 077635 763414 609472 306985 509348 665816 455585 305105 584465 693533 550563 478028 900530 419583 122893 044409 342553 148802 184479 355370 407144 690513 587582 412636 566323 400530 693953 041966 113720 932344 853810 977116 709455 520545 562392 926847 129446 607845 046145 785652 624457 189729 350396 065779 248691 278678 564967 916483 490586 466297 471904 577323 283756 984338 934834 588959 918878 690266 147144 743607 221422 254167 368645 800011 408773 852850 297811 849216 960855 780719 066591 702182 202324 227437 945200 504992 432579 975703 317082 808013 270624 218635 696122 249155 180152 187143 551386 223433 948731 416743 126652 377662 918583 927319 881617 392867 764081 800694 531374 801608 564646 652263 560058 013511 308301 594991 305888 585319 445835 698330 134802 294455 957406 877963 585177 685538 750163 386144 663461 522849 415415 361118 541022 253936 445257 234674 922421 773824 120645 956927 436934 246452 242304 333696 903932 556296 688144 917365 999602 189145 168675 930407 484731 261527 588271 651650 220437 012091 865651 022049 782071 989152 194236 368178 088641 910641 < 251213 [i]
- extracting embedded orthogonal array [i] would yield linear OA(25105, 1318, F25, 90) (dual of [1318, 1213, 91]-code), but
(106−91, 106, 1318)-Net in Base 25 — Upper bound on s
There is no (15, 106, 1319)-net in base 25, because
- 1 times m-reduction [i] would yield (15, 105, 1319)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 619 328459 230983 605141 367126 508049 418867 300979 629693 946130 236958 321791 016988 299579 362475 825487 531914 768145 401519 852476 123717 606852 112658 994241 934985 > 25105 [i]