MinT - Best Known (42, 145, s)-Nets in Base 4

Best Known (42, 145, s)-Nets in Base 4

(42, 145, 56)-Net over F₄ — Constructive and digital

Digital (42, 145, 56)-net over F₄, using

t-expansion [i] based on digital (33, 145, 56)-net over F₄, using
- net from sequence [i] based on digital (33, 55)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ 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]

(42, 145, 75)-Net over F₄ — Digital

Digital (42, 145, 75)-net over F₄, using

t-expansion [i] based on digital (40, 145, 75)-net over F₄, using
- net from sequence [i] based on digital (40, 74)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 40 and N(F) ≥ 75, using
    - Drinfeld modules of rank 1 [i]

(42, 145, 290)-Net in Base 4 — Upper bound on s

There is no (42, 145, 291)-net in base 4, because

1 times m-reduction [i] would yield (42, 144, 291)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4¹⁴⁴, 291, S₄, 102), but
  - 9 times code embedding in larger space [i] would yield OA(4¹⁵³, 300, S₄, 102), but
    - the linear programming bound shows that MÂ â‰¥ 399 404342 388942 094544 744509 831104 667890 936998 605192 130964 918812 017381 737451 016379 393595 721379 101359 308088 877121 362116 659171 080724 407185 500975 841643 398925 402713 336082 828659 050981 061746 982915 553913 637665 286237 999580 812132 064412 155466 648327 705073 495169 606936 988697 495469 345672 031085 185548 298741 545690 508415 300239 046190 522917 620333 658094 795552 734289 617473 491621 781366 539111 731594 097523 088287 517116 990136 854747 876601 026143 714084 360091 674196 398557 565570 637398 464599 352945 129876 977280 507524 900934 426062 705094 840540 905046 601623 588731 558407 098248 398984 766324 369866 346333 457424 940193 002167 645334 339593 070478 261881 149271 677185 337442 666676 036137 815729 748158 939689 759514 301349 989820 033781 918184 057678 663057 399995 454954 893149 265093 140636 141844 817639 389842 022484 198545 636826 046093 730773 081267 995530 451083 416103 228918 256731 226800 302916 261934 451792 655080 817471 824006 880532 273136 102588 426140 640202 025230 217273 143745 369735 307408 752042 345566 981072 631220 548676 340404 875024 370581 904366 379175 109618 689666 096619 667138 490549 395810 605711 757102 593647 091254 956163 939856 460779 531508 618002 685093 598323 236872 633867 080964 571872 907026 067577 602878 121789 777603 760927 327868 857933 278268 053950 996614 160138 392767 374183 042901 202017 755740 245354 921490 300069 454439 190055 231301 095279 389467 614054 243323 109748 322290 824233 513788 563037 625695 496352 248185 823798 180895 634941 163048 155737 096503 637147 870449 246290 598376 567685 128155 383108 345591 234560 / 2 038934 054244 935999 742869 972399 131654 787581 138580 884579 506503 772347 731124 856408 824990 480979 621092 144696 154791 166097 339260 040035 097675 247933 979234 266321 024050 631456 930544 708753 085542 048359 096217 853617 392959 588238 516605 235818 127031 734854 524678 014628 079329 260176 963155 030929 554169 900115 393067 465705 855414 342752 091720 132066 025660 057140 352491 075110 127474 205420 054514 991199 306926 638394 337282 863593 959149 155623 886270 145567 829739 202835 657567 361760 580268 453308 786920 631744 337884 630042 796188 106662 618011 259715 101854 620070 673213 160424 400936 496643 477329 713228 525134 010768 892929 053503 648866 077765 592494 270822 893932 138848 500496 036837 213508 773956 032605 469790 174299 649749 966385 961712 159906 418737 751505 008683 314147 261215 793184 094326 753431 145515 532480 523228 635969 169830 097107 477917 334158 150988 452233 906575 865821 375707 629690 491506 624682 050922 530614 081315 327604 779029 779025 161454 528177 570315 909941 602732 485527 473463 364581 880804 006429 669692 147773 211815 082339 090893 762280 655670 356859 128270 992242 156339 328823 111398 620038 246859 621609 961022 862663 778814 361537 450185 919049 771469 489602 372983 655561 924492 977345 181321 123630 417801 496505 618685 839431 709293 477086 583295 502133 144482 658146 213338 305274 061924 955480 088406 598322 792257 877907 304837 359106 566049 524761 240411 934102 712088 276462 472830 091793 882775 491331 403561 483245 824440 122453 810839 > 4¹⁵³ [i]

Best Known (42, 145, s)-Nets in Base 4

(42, 145, 56)-Net over F4 — Constructive and digital

(42, 145, 75)-Net over F4 — Digital

(42, 145, 290)-Net in Base 4 — Upper bound on s

(42, 145, 56)-Net over F₄ — Constructive and digital

(42, 145, 75)-Net over F₄ — Digital