Best Known (109−97, 109, s)-Nets in Base 25
(109−97, 109, 126)-Net over F25 — Constructive and digital
Digital (12, 109, 126)-net over F25, using
- t-expansion [i] based on digital (10, 109, 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
(109−97, 109, 1055)-Net over F25 — Upper bound on s (digital)
There is no digital (12, 109, 1056)-net over F25, because
- 17 times m-reduction [i] would yield digital (12, 92, 1056)-net over F25, but
- extracting embedded orthogonal array [i] would yield linear OA(2592, 1056, F25, 80) (dual of [1056, 964, 81]-code), but
- the Johnson bound shows that N ≤ 4101 215677 043668 004788 442252 447194 682379 645842 639674 456481 602824 102025 188158 127658 851977 439601 863978 236418 837062 269321 451652 342592 351808 010908 217002 889834 331738 038584 151057 541068 940416 791264 770362 144255 007448 514859 519494 788802 102080 222806 026058 025038 313528 056320 415607 534987 478884 012792 386972 900159 134740 661564 152711 202217 459561 864231 579325 238751 129588 713631 360988 388704 600722 368999 397756 734138 413028 520043 937247 999503 096906 001690 804777 952403 816243 355655 018283 925497 808127 610009 256894 199145 888288 837300 504118 537372 802643 110965 240006 555628 600141 540588 953472 617093 574456 403639 840240 211792 562738 028969 380285 681269 696947 162236 193932 449960 869218 459024 042752 758630 387389 605590 699306 945453 113610 632285 549595 558249 850414 570803 089940 192484 959810 087011 572723 425821 633735 093018 505074 893821 301584 525875 144366 744250 589334 998338 764406 547733 831210 800257 557222 905325 153529 288586 631936 860438 634278 221620 738490 695014 823810 002419 656536 860324 630748 173310 542991 600318 804090 356623 694119 020783 768254 697813 122948 364583 398761 561011 523298 742149 122596 223193 294787 442260 856912 384464 780494 853824 659605 677164 706290 518086 824900 279943 758763 911683 447075 580315 642820 688795 457259 341583 020720 341539 295815 824471 142958 347333 446851 623665 169564 825285 768872 570819 896148 852098 937960 261061 786471 998201 442797 245634 513490 144356 847307 161559 654360 690448 221093 845874 310848 970414 908300 769786 139587 < 25964 [i]
- extracting embedded orthogonal array [i] would yield linear OA(2592, 1056, F25, 80) (dual of [1056, 964, 81]-code), but
(109−97, 109, 1056)-Net in Base 25 — Upper bound on s
There is no (12, 109, 1057)-net in base 25, because
- 11 times m-reduction [i] would yield (12, 98, 1057)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 99931 553995 410600 915510 329562 529837 879229 429754 477829 461184 370325 276741 690311 479173 064112 823773 171335 589274 557819 242658 121926 099542 849225 > 2598 [i]