MinT - Best Known (144−99, 144, s)-Nets in Base 4

Best Known (144−99, 144, s)-Nets in Base 4

(144−99, 144, 56)-Net over F₄ — Constructive and digital

Digital (45, 144, 56)-net over F₄, using

t-expansion [i] based on digital (33, 144, 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]

(144−99, 144, 80)-Net over F₄ — Digital

Digital (45, 144, 80)-net over F₄, using

net from sequence [i] based on digital (45, 79)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 45 and N(F) ≥ 80, using
  - Drinfeld modules of rank 1 [i]

(144−99, 144, 294)-Net in Base 4 — Upper bound on s

There is no (45, 144, 295)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4¹⁴⁴, 295, S₄, 99), but
- 5 times code embedding in larger space [i] would yield OA(4¹⁴⁹, 300, S₄, 99), but
  - the linear programming bound shows that MÂ â‰¥ 57 971682 944249 904570 760685 174961 482704 028158 090645 460751 517687 857828 332865 127797 351038 989627 833289 132480 799447 539780 383770 891344 553603 834835 901830 395562 806663 567017 055894 686631 935338 805307 123117 736830 071540 218617 174055 121358 642492 185740 454580 646274 413883 033458 105940 977914 398770 304884 236675 508801 370762 088916 067512 466914 141985 323871 721807 176095 999012 725181 835589 020653 579766 330468 621299 703769 681863 116626 001742 796161 642402 816676 332076 373583 217768 879759 580482 315159 040598 497932 735657 255961 534077 821051 749146 687249 519894 717132 529061 753151 085323 357197 826292 110143 215986 163780 307056 240994 698498 929076 266415 711967 498539 217530 289907 292696 199184 216793 111554 161870 691141 467779 826937 323680 039683 352399 842779 445993 358009 949615 888535 767375 880522 984216 020990 657121 169531 938821 734827 130409 473306 995292 772353 381916 972798 159613 483149 760687 714740 329603 773997 391925 372197 543668 358308 052518 266182 425610 070460 191778 319682 905453 866895 251297 434376 179620 866056 224765 734545 659011 245579 544620 940793 172051 037922 038505 676254 011342 385104 211105 373608 868515 834101 147826 925052 190218 478476 710499 197450 828327 171868 030271 275078 287777 138549 606768 635002 026484 699302 700942 343192 063403 646491 055418 839067 962173 257434 350733 326511 171555 254644 769659 087390 403475 884734 675507 492206 955013 039168 956095 135200 379973 766254 998150 723718 229135 449310 630501 581901 835291 576229 619266 854994 428008 318071 972305 362754 400538 284357 347487 255520 683206 332897 404752 492914 232811 801761 282578 436185 246018 341749 211069 019366 779221 911846 023306 852061 454442 922056 802751 926906 051349 456647 399382 717645 337310 858674 013623 421533 747107 956584 528598 747349 189868 688838 551509 983457 561842 312546 644979 840758 841344 / 97 871155 164773 372788 354538 307053 546948 572024 150153 284964 867644 985720 898547 589161 330913 819151 217326 473279 816969 895311 925936 503388 036510 444302 475806 989131 416712 122463 521555 381516 236367 916583 512751 580717 602632 862670 883095 513407 555784 223218 753284 994135 701844 744682 184353 268785 506784 087797 495798 820634 543255 639559 687321 360205 101065 114480 454020 974906 667763 689525 865113 081899 906887 674275 308303 599014 674392 462877 251642 885830 335321 584347 048369 404495 086423 195713 788781 247436 453140 247187 621908 287319 885643 220617 424240 828398 991310 103731 225781 084409 208995 745596 500240 852971 616422 665840 459563 989272 187612 782207 436878 094697 374147 935243 179056 639359 803286 960655 561939 831590 613443 972467 262954 749923 948297 466944 178806 553659 655603 924244 760398 333503 252750 504283 484737 074253 295147 285669 185765 468304 318871 425738 683437 478526 333762 956794 170040 812103 720592 620602 939577 224847 789407 325389 565240 074891 815842 355903 913460 069227 615638 499784 139009 436268 421312 146550 118994 347413 859111 950678 260660 337806 950468 429586 696117 258948 918331 692933 767497 138948 716807 523770 612457 021773 097374 999390 186196 534485 571080 380137 702492 805064 420260 574451 130218 343962 187723 122177 879317 267259 141189 495308 388942 119983 680121 803850 854014 074683 774190 360773 442925 641090 060169 725188 866357 429536 528188 170867 605835 741642 484499 919919 503613 374477 754395 533760 088719 202874 923147 375077 494319 671695 460911 869793 650411 511670 744210 714997 036157 687318 213799 500798 872983 186836 747563 088989 893132 255219 807710 160841 275932 356081 326499 569866 170358 996295 647844 458606 596556 952927 879872 821152 789860 395254 482710 366548 669547 > 4¹⁴⁹ [i]

Best Known (144−99, 144, s)-Nets in Base 4

(144−99, 144, 56)-Net over F4 — Constructive and digital

(144−99, 144, 80)-Net over F4 — Digital

(144−99, 144, 294)-Net in Base 4 — Upper bound on s

(144−99, 144, 56)-Net over F₄ — Constructive and digital

(144−99, 144, 80)-Net over F₄ — Digital