Best Known (82−72, 82, s)-Nets in Base 25
(82−72, 82, 126)-Net over F25 — Constructive and digital
Digital (10, 82, 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
(82−72, 82, 888)-Net over F25 — Upper bound on s (digital)
There is no digital (10, 82, 889)-net over F25, because
- 2 times m-reduction [i] would yield digital (10, 80, 889)-net over F25, but
- extracting embedded orthogonal array [i] would yield linear OA(2580, 889, F25, 70) (dual of [889, 809, 71]-code), but
- the Johnson bound shows that N ≤ 843 136687 935573 355636 591238 518473 494087 332590 788839 022862 737319 269888 725944 800379 624617 175215 562704 444834 306304 855902 063362 734936 764722 898682 415113 666071 138535 657446 372184 840753 456343 305818 313088 508524 078813 677526 979769 196926 322024 466295 559465 661299 800206 369478 213220 947838 933754 204761 010286 671464 801163 740231 829145 234042 310904 421805 184457 312855 443932 049662 451711 281550 654839 456272 237319 055067 812419 071014 805273 154400 147659 249097 970084 038454 357260 643528 912736 530800 436251 692852 521190 556001 018296 627160 973844 566859 933366 155665 436138 869005 118536 053010 793801 898397 309017 667508 571903 938705 215091 557614 100782 299256 410024 789777 458567 848411 023307 192817 095226 634034 458339 807768 211986 373999 610007 577275 859773 775189 199274 502334 426794 353579 950286 638264 445830 809934 548772 596082 981472 844587 297789 417466 696373 319164 546312 873599 066950 709991 781605 705382 180539 690920 826085 750100 594036 993234 349960 086352 093393 935586 507238 448782 947525 166868 998146 453930 713531 771801 681489 376964 267106 335107 222135 493479 672932 201810 596190 966384 243801 054336 395691 876965 168174 940226 195958 993174 806479 457361 106047 338905 520592 919551 708344 893666 708155 108446 144425 786394 838462 < 25809 [i]
- extracting embedded orthogonal array [i] would yield linear OA(2580, 889, F25, 70) (dual of [889, 809, 71]-code), but
(82−72, 82, 890)-Net in Base 25 — Upper bound on s
There is no (10, 82, 891)-net in base 25, because
- the generalized Rao bound for nets shows that 25m ≥ 4 375536 351548 438605 882641 751894 190128 907859 387788 444341 979002 877673 988783 548365 981729 750173 701668 539416 699723 669025 > 2582 [i]