Best Known (42, 42+104, s)-Nets in Base 4
(42, 42+104, 56)-Net over F4 — Constructive and digital
Digital (42, 146, 56)-net over F4, using
- t-expansion [i] based on digital (33, 146, 56)-net over F4, using
- net from sequence [i] based on digital (33, 55)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 33 and N(F) ≥ 56, using
- F5 from the tower of function fields by GarcÃa and Stichtenoth over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 33 and N(F) ≥ 56, using
- net from sequence [i] based on digital (33, 55)-sequence over F4, using
(42, 42+104, 75)-Net over F4 — Digital
Digital (42, 146, 75)-net over F4, using
- t-expansion [i] based on digital (40, 146, 75)-net over F4, using
- net from sequence [i] based on digital (40, 74)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 40 and N(F) ≥ 75, using
- net from sequence [i] based on digital (40, 74)-sequence over F4, using
(42, 42+104, 276)-Net over F4 — Upper bound on s (digital)
There is no digital (42, 146, 277)-net over F4, because
- extracting embedded orthogonal array [i] would yield linear OA(4146, 277, F4, 104) (dual of [277, 131, 105]-code), but
- residual code [i] would yield OA(442, 172, S4, 26), but
- the linear programming bound shows that M ≥ 34 487674 602431 075505 978819 529968 455340 679044 792320 / 1 686703 652728 844301 656717 > 442 [i]
- residual code [i] would yield OA(442, 172, S4, 26), but
(42, 42+104, 289)-Net in Base 4 — Upper bound on s
There is no (42, 146, 290)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 8397 114472 259945 659095 982979 773076 104418 530856 975321 822400 650976 322451 667192 054981 375040 > 4146 [i]
- extracting embedded orthogonal array [i] would yield OA(4146, 290, S4, 104), but
- 10 times code embedding in larger space [i] would yield OA(4156, 300, S4, 104), but
- the linear programming bound shows that M ≥ 152 174367 376618 948761 331162 024662 260438 663355 968520 799998 749018 272316 566002 218384 586843 722883 998947 548768 897232 570992 593012 791560 021125 962406 513359 922901 705637 804149 705522 399099 248311 993034 845392 399705 653689 984172 633066 290214 780464 822037 365033 542706 587568 302848 564033 598478 278402 391542 844626 690356 444670 206210 562250 153614 053561 551403 172445 637274 566141 868305 017276 569685 924386 699904 968504 172104 749286 767367 244849 784250 109931 706316 286392 386012 059509 348232 167170 013503 369544 927371 453224 948665 345177 681023 706858 665091 726066 903765 226488 867951 381805 867017 258716 677257 866672 153578 362353 277635 200725 051563 616864 783225 359871 281024 031935 971238 126355 597183 816038 053427 254790 849958 629395 312641 665065 514662 348096 574121 162881 221499 780751 240144 574329 022152 205440 660125 013171 623327 839800 172999 159352 752799 412245 412503 640194 061542 472072 922553 917319 357189 067857 553917 170974 112817 157185 001301 331223 014877 049936 692317 750531 806917 391424 933069 637336 242604 244635 438927 320080 547030 876870 585005 821006 802285 247298 613547 783636 901245 513190 097340 989956 844084 657425 225277 991005 122555 133685 581921 177436 972330 618587 406141 489426 104686 859278 749077 735167 673235 607906 156544 / 16453 547584 992501 620365 659783 936494 755476 838331 238570 570917 955466 110710 088382 557968 108158 222920 633839 455198 339252 897498 892197 144469 708089 169260 406133 154296 517562 129379 745177 226762 354820 158143 347330 467095 372535 268784 021089 382739 711498 662038 885292 677060 796906 063171 534156 155049 631000 913767 866569 551816 171560 884840 247927 772151 662079 905092 585821 335084 147990 733518 785142 279283 421288 827154 942684 536454 124270 712577 613736 170425 736275 610558 293448 672993 116994 982756 958521 944097 135313 442919 007171 702822 650842 073687 935184 112295 355387 580768 118942 569211 905655 205502 962105 890093 979564 596355 050704 480590 636406 816834 043976 987321 853390 167664 096670 971428 783595 821152 358772 797185 289298 897966 707893 163393 225123 262911 548634 502097 808044 406741 469724 741733 190994 365909 782479 380914 702585 928621 525068 595838 751047 101661 628041 765178 566899 468728 050044 424774 167276 153367 073691 767534 995787 479145 379161 038608 066248 194411 535908 013928 034905 135194 824785 832669 697170 474281 163608 833419 042624 204799 977227 183264 396696 858980 187743 293139 599184 822382 413932 884749 180201 034867 180117 191875 > 4156 [i]
- 10 times code embedding in larger space [i] would yield OA(4156, 300, S4, 104), but