MinT - Best Known (42, 42+96, s)-Nets in Base 4

Best Known (42, 42+96, s)-Nets in Base 4

(42, 42+96, 56)-Net over F₄ — Constructive and digital

Digital (42, 138, 56)-net over F₄, using

t-expansion [i] based on digital (33, 138, 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, 42+96, 75)-Net over F₄ — Digital

Digital (42, 138, 75)-net over F₄, using

t-expansion [i] based on digital (40, 138, 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, 42+96, 293)-Net in Base 4 — Upper bound on s

There is no (42, 138, 294)-net in base 4, because

2 times m-reduction [i] would yield (42, 136, 294)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4¹³⁶, 294, S₄, 94), but
  - 6 times code embedding in larger space [i] would yield OA(4¹⁴², 300, S₄, 94), but
    - the linear programming bound shows that MÂ â‰¥ 9 153888 328887 476977 714307 710865 674461 239896 794713 412673 457711 977185 417601 730447 992827 683638 417499 523667 813997 688623 997643 329795 159711 134389 133429 807923 870866 515214 682846 584052 432377 192375 340679 284479 581820 819426 715354 509523 216694 723903 451482 290922 892959 490101 136759 356137 681897 201750 665139 738078 585395 874889 089272 224743 587813 987584 696722 653694 917357 402281 969879 751357 718757 915744 363082 942088 378343 793762 613071 745256 359514 191137 918155 137429 854942 482684 190738 672272 323734 494401 660878 990491 773097 636088 020286 819818 015877 362362 961643 848049 016293 137007 849192 866083 713758 284785 217107 791881 674979 095585 942019 682895 123920 870492 482341 312634 349869 350486 343091 830426 724928 174502 322377 801339 311937 569993 527372 312134 936166 764696 584357 684378 001221 241681 377036 272514 116653 892303 333692 522158 568359 756942 924311 777173 949493 057268 518504 171554 744576 224826 035310 742861 825711 188333 220116 500115 786538 161381 555788 666715 627238 229210 975212 404736 / 204328 848734 117724 934216 724446 405178 908763 769050 316332 789077 812013 467394 402707 972830 983001 897292 836503 317185 374416 558449 717899 858986 601315 542000 403792 638040 957542 371628 046204 130984 884607 486748 992833 062723 351309 422780 293967 796144 989899 216645 193007 616654 453300 885701 856887 833414 547943 950745 510394 649684 150673 948959 457926 275326 009551 598804 250670 061364 937610 381807 104105 897860 802247 241454 100440 864756 083240 608727 547446 100417 923847 889750 626288 061313 181786 751057 839507 078509 484070 223297 134140 568222 225833 888370 732440 281107 465300 901449 457086 404505 621271 745253 223924 163253 487051 055237 587226 154626 080742 597168 398421 956895 235990 670420 942073 021149 876475 305626 400814 137728 355245 844035 340640 885050 479249 845706 593007 795476 326282 270554 720229 515228 284153 648390 936847 481476 333629 384409 082494 963742 000168 490973 995830 942221 427554 885433 733342 206866 994939 > 4¹⁴² [i]

Best Known (42, 42+96, s)-Nets in Base 4

(42, 42+96, 56)-Net over F4 — Constructive and digital

(42, 42+96, 75)-Net over F4 — Digital

(42, 42+96, 293)-Net in Base 4 — Upper bound on s

(42, 42+96, 56)-Net over F₄ — Constructive and digital

(42, 42+96, 75)-Net over F₄ — Digital