Best Known (131−100, 131, s)-Nets in Base 5
(131−100, 131, 72)-Net over F5 — Constructive and digital
Digital (31, 131, 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
(131−100, 131, 270)-Net over F5 — Upper bound on s (digital)
There is no digital (31, 131, 271)-net over F5, because
- 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
(131−100, 131, 292)-Net in Base 5 — Upper bound on s
There is no (31, 131, 293)-net in base 5, because
- 3 times m-reduction [i] would yield (31, 128, 293)-net in base 5, but
- extracting embedded orthogonal array [i] would yield OA(5128, 293, S5, 97), but
- 7 times code embedding in larger space [i] would yield OA(5135, 300, S5, 97), but
- the linear programming bound shows that M ≥ 12852 167659 703928 755848 434925 135762 258877 531299 034780 656775 530464 909476 905959 108001 466932 501301 079685 459430 482886 602489 289710 076040 838654 464692 724834 187882 332650 434657 596611 446839 453870 366500 741361 091216 599586 187749 744512 586092 504202 701901 185541 978910 013482 684163 838984 122779 229891 404910 307229 575474 292473 819158 398418 392385 353377 411620 903812 116291 772033 955551 751135 851261 187941 859459 086365 541866 082018 713652 366523 491074 570667 430170 281215 953673 922596 410736 551615 768727 568218 893675 122593 457250 410949 356332 927384 591256 042062 291752 465480 758640 545677 855551 704747 900385 803510 683839 760583 254262 979695 295133 144443 681832 422039 876935 394038 691642 279326 282758 635857 156616 944158 868782 060819 963920 644684 267714 155682 112412 773134 757166 351490 820000 306260 175014 628969 018782 652879 308660 046315 929316 603156 557812 159160 981570 028481 346645 249681 333226 686629 388485 148276 432493 835708 115170 746168 587356 805801 391601 562500 / 557232 488410 562391 762391 288922 689841 201645 485063 175287 598170 667542 739293 014328 218750 487095 794223 533657 099519 198578 290430 641027 697373 365195 429369 895401 486012 290515 687253 681743 398805 754532 005873 713515 709295 485734 410329 127176 180744 751035 526034 340493 696360 728114 626047 790281 665342 982204 467554 626733 711539 730820 103522 046908 057107 646471 835523 050818 178766 902520 988838 075021 040599 083307 042886 921031 533689 748439 499788 439105 857836 800964 465474 356559 767835 297747 520021 761959 109840 767569 826772 006192 959941 995591 788912 303775 071234 794180 952771 965822 140490 278110 534974 402394 452543 196127 052703 757745 352811 185062 083976 751208 813487 274297 159188 407661 972777 544536 864237 048431 554195 640698 787576 251709 623159 617779 449198 615069 431118 739105 958256 355657 513952 518742 815813 972333 612878 018616 002408 323347 244623 951410 572189 588463 > 5135 [i]
- 7 times code embedding in larger space [i] would yield OA(5135, 300, S5, 97), but
- extracting embedded orthogonal array [i] would yield OA(5128, 293, S5, 97), but