Best Known (109−98, 109, s)-Nets in Base 25
(109−98, 109, 126)-Net over F25 — Constructive and digital
Digital (11, 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
(109−98, 109, 971)-Net over F25 — Upper bound on s (digital)
There is no digital (11, 109, 972)-net over F25, because
- 22 times m-reduction [i] would yield digital (11, 87, 972)-net over F25, but
- extracting embedded orthogonal array [i] would yield linear OA(2587, 972, F25, 76) (dual of [972, 885, 77]-code), but
- the Johnson bound shows that N ≤ 14 887648 240835 464713 684001 494543 827853 825154 886726 537182 227428 743210 490267 256238 121737 830708 709230 479115 304833 312141 339077 215394 777094 207059 205011 646336 236568 184021 519824 811133 755958 879467 949929 027913 904348 215471 615352 887143 359116 356020 197570 251469 513809 271445 665000 065932 749689 877669 924035 281897 662756 357863 753398 702022 105900 282641 843363 833698 520483 559514 868279 858839 177022 805914 525306 853479 171301 247014 519669 347027 763935 896449 509195 856859 493694 501424 096719 202784 276388 170618 010773 349948 117310 336133 315189 257012 884350 176405 459163 942290 489033 310326 175820 012888 784795 577033 348056 277812 598627 934084 867296 974672 690194 618284 776315 948442 273650 270030 479118 905261 358725 858669 598591 875711 396115 322536 921328 158646 738960 992921 662119 871791 749016 010399 228832 579355 254260 286185 396833 658888 956268 140293 517091 401374 219802 356100 192275 524896 783900 164621 657129 340606 447647 699705 320750 033179 324402 791649 620517 333407 172737 954629 406024 576381 146811 409053 196301 938206 807753 875950 650343 138138 230265 258730 296807 228322 733110 166609 386476 565696 887292 495677 246534 649946 172170 904908 947608 028994 095559 043196 935342 213649 268738 497079 632026 579986 125095 260309 338726 935869 456677 780922 416016 728415 892394 181668 247923 555391 409077 463074 658962 745522 792842 844012 022252 735914 669062 < 25885 [i]
- extracting embedded orthogonal array [i] would yield linear OA(2587, 972, F25, 76) (dual of [972, 885, 77]-code), but
(109−98, 109, 973)-Net in Base 25 — Upper bound on s
There is no (11, 109, 974)-net in base 25, because
- 20 times m-reduction [i] would yield (11, 89, 974)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 26652 679429 255650 895205 725392 730878 485909 592073 299875 632424 536330 186889 064812 117998 496443 018580 423215 484992 814849 507886 536305 > 2589 [i]