Best Known (13, 13+87, s)-Nets in Base 25
(13, 13+87, 126)-Net over F25 — Constructive and digital
Digital (13, 100, 126)-net over F25, using
- t-expansion [i] based on digital (10, 100, 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
(13, 13+87, 1139)-Net over F25 — Upper bound on s (digital)
There is no digital (13, 100, 1140)-net over F25, because
- 1 times m-reduction [i] would yield digital (13, 99, 1140)-net over F25, but
- extracting embedded orthogonal array [i] would yield linear OA(2599, 1140, F25, 86) (dual of [1140, 1041, 87]-code), but
- the Johnson bound shows that N ≤ 1794 363156 412127 221347 197098 423749 352535 039382 616176 254060 873021 894278 823094 531205 758288 397577 081726 882869 665403 442776 081650 742352 950582 880428 122249 221928 874648 055551 603299 261497 542372 653423 478081 483087 437086 275833 802161 362180 233341 324942 578233 829225 949409 701175 058634 161165 265052 669914 255923 005875 563074 167947 001864 496046 091126 155266 595163 534195 581440 956410 309129 293019 242642 083725 527306 797555 011512 127082 616624 205725 249834 754473 419353 144742 953335 487725 485551 108070 172990 257553 739968 923497 047549 377362 010493 125559 312717 286597 127360 637171 599846 394690 102857 695744 560247 483479 556834 092229 561058 901305 795504 149577 027579 182035 592039 250678 351374 099731 767426 453419 887870 638468 847815 694915 798236 522280 688603 933681 757181 834619 469117 912779 014703 076383 246747 279487 523008 466534 483298 892797 164837 446521 698172 868808 217072 189522 695003 251131 469629 824337 398650 131494 161068 017332 033146 661687 712537 267835 027969 670677 401522 811938 766842 586257 405088 261601 005701 540312 100347 668744 367522 749861 350391 676445 077277 493687 024669 366655 149238 223315 386732 101961 521628 849621 131052 987886 261551 923135 410129 411734 684842 321328 944065 955645 612556 668558 374586 030834 702577 242004 017651 356953 757642 078770 339970 675382 586662 164140 937425 321588 237938 148340 562786 848394 973733 578868 806080 563055 497534 658715 879621 714596 106140 857615 508715 091945 996303 325952 919995 323495 636306 613203 600950 501191 137209 681842 504339 782570 969875 407012 924154 616058 064733 389903 288936 362307 355232 602499 552372 667845 982676 043817 696300 < 251041 [i]
- extracting embedded orthogonal array [i] would yield linear OA(2599, 1140, F25, 86) (dual of [1140, 1041, 87]-code), but
(13, 13+87, 1140)-Net in Base 25 — Upper bound on s
There is no (13, 100, 1141)-net in base 25, because
- 1 times m-reduction [i] would yield (13, 99, 1141)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 2 505974 551751 515068 927267 255124 805712 614402 234390 461025 587167 867613 994341 256544 510574 819519 347506 063538 155217 121382 008294 943216 731908 891625 > 2599 [i]