Best Known (31, 31+104, s)-Nets in Base 5
(31, 31+104, 72)-Net over F5 — Constructive and digital
Digital (31, 135, 72)-net over F5, using
- net from sequence [i] based on digital (31, 71)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 31 and N(F) ≥ 72, using
(31, 31+104, 270)-Net over F5 — Upper bound on s (digital)
There is no digital (31, 135, 271)-net over F5, because
- 4 times m-reduction [i] would yield digital (31, 131, 271)-net over F5, but
- extracting embedded orthogonal array [i] would yield linear OA(5131, 271, F5, 100) (dual of [271, 140, 101]-code), but
- residual code [i] would yield OA(531, 170, S5, 20), but
- the linear programming bound shows that M ≥ 42216 069885 091800 891878 320552 825927 734375 / 8 765861 509813 289221 > 531 [i]
- residual code [i] would yield OA(531, 170, S5, 20), but
- extracting embedded orthogonal array [i] would yield linear OA(5131, 271, F5, 100) (dual of [271, 140, 101]-code), but
(31, 31+104, 291)-Net in Base 5 — Upper bound on s
There is no (31, 135, 292)-net in base 5, because
- 3 times m-reduction [i] would yield (31, 132, 292)-net in base 5, but
- extracting embedded orthogonal array [i] would yield OA(5132, 292, S5, 101), but
- 8 times code embedding in larger space [i] would yield OA(5140, 300, S5, 101), but
- the linear programming bound shows that M ≥ 152 242666 725366 631530 851383 795970 562527 269210 631927 869607 860781 401107 826384 405562 293721 712624 507446 443840 974014 045791 087055 370426 329638 420117 203013 731767 571100 513787 290086 621274 378176 245180 456212 405838 763347 976335 950107 536549 799336 427267 963074 610791 240120 843971 180845 703135 596546 426795 120230 885124 596503 974630 936749 727485 255275 956903 504391 128617 714876 409828 131640 573520 311152 317774 182678 787528 870390 387871 524642 839621 222131 088369 216188 809028 138522 652590 468228 821638 670483 593342 299479 320158 413712 128300 304676 960956 925407 125837 188558 673614 417661 099189 734950 806616 919532 978514 044125 154004 438964 891619 214960 384896 346569 906953 319154 119207 819842 099835 419236 097529 635642 631600 014342 277181 884448 193075 157962 149257 171814 728046 003451 187109 771865 773467 624907 597207 697054 774163 854584 985577 882695 242756 009638 090071 674899 591266 990880 210161 984789 005999 444415 598034 233635 117351 953627 345021 592208 325992 991594 116691 527465 109108 719255 579453 631767 951637 697470 182434 139953 030449 576938 525298 530192 123750 057098 611428 587095 177096 488316 499327 858219 957343 433037 522117 956541 478633 880615 234375 / 1 713607 934588 629711 782194 704913 502003 816841 146743 026988 216719 925327 527397 380455 950199 592707 023330 080796 358057 175996 445528 303284 654439 459360 633325 588947 843284 910386 538967 405364 664580 033084 343199 062391 824705 670085 945586 785852 752681 685598 853706 972577 459822 837085 602695 649758 113976 961135 232670 540909 994413 952468 080458 867075 058478 481013 423099 481144 233085 724393 841895 236761 458891 065052 389245 105995 221475 730496 587566 737429 832011 768466 820651 630805 138212 180892 389177 706288 834723 396909 494733 120835 880750 243975 642590 675520 852542 129838 156511 230915 318944 887024 010321 113424 332754 449053 313728 595058 187986 346977 239680 603536 413657 137065 624020 352582 574734 670526 565977 910697 293224 254226 059945 280504 264212 496194 220545 389467 289095 950961 672389 244877 708005 539990 931098 806941 884538 140895 774606 529512 862407 203199 393609 818189 391702 523468 629868 472603 836425 194433 279947 313027 476962 232994 103969 510786 179748 012474 282216 789841 548887 724533 828030 546990 729092 784397 773287 548603 099479 057902 989769 961053 > 5140 [i]
- 8 times code embedding in larger space [i] would yield OA(5140, 300, S5, 101), but
- extracting embedded orthogonal array [i] would yield OA(5132, 292, S5, 101), but