MinT - Best Known (42, 42+91, s)-Nets in Base 4

Best Known (42, 42+91, s)-Nets in Base 4

(42, 42+91, 56)-Net over F₄ — Constructive and digital

Digital (42, 133, 56)-net over F₄, using

t-expansion [i] based on digital (33, 133, 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+91, 75)-Net over F₄ — Digital

Digital (42, 133, 75)-net over F₄, using

t-expansion [i] based on digital (40, 133, 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+91, 294)-Net in Base 4 — Upper bound on s

There is no (42, 133, 295)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4¹³³, 295, S₄, 91), but
- 5 times code embedding in larger space [i] would yield OA(4¹³⁸, 300, S₄, 91), but
  - the linear programming bound shows that MÂ â‰¥ 23 606356 966186 468840 422483 274755 892711 515095 056514 764625 571895 924005 255783 480525 515191 291442 849200 936336 271249 519641 697917 230710 915812 910201 079018 129685 675728 819684 633021 578574 039410 080756 265810 463574 695007 690931 764015 070040 704141 386222 531030 084826 962669 014485 473085 330926 574871 661610 085731 848804 200687 084124 920843 106992 644072 960159 649670 735073 817871 884952 436422 571940 112669 267263 754359 681810 347626 682630 447349 232097 398122 691186 177490 663734 851293 905611 708608 240571 359850 024953 659915 268520 943875 191461 185216 118403 749583 643659 750092 138318 819898 614979 900871 064592 648216 923643 386278 693906 719960 315848 611379 030018 595781 209125 171297 279353 592214 492839 772166 504686 580703 493614 143143 142033 302247 596888 345742 373747 889963 205123 378846 086008 510996 262592 794247 578929 711624 261138 039686 823936 / 158 250011 269963 368682 265849 517431 433028 053304 119132 701461 662735 762474 956229 976822 108535 830924 088887 789226 038581 463728 909228 521449 120334 452205 486055 702892 879243 429428 898490 281987 251642 628753 923350 310177 613995 539446 976428 462271 493561 732699 478982 391935 753017 373485 194672 680644 872783 613993 470080 788371 848155 412733 398716 305759 186881 101846 398964 929990 134720 656497 145095 395026 821754 307217 774226 171931 475903 015009 083332 030142 267975 152772 361664 943233 270523 964988 027619 157127 724409 514860 286803 200042 741026 743909 404876 489903 376009 190985 607919 480419 770964 358200 492924 880474 601755 572533 300811 247320 904900 251167 687839 313636 407695 356582 292132 782753 228731 631225 388525 845235 678692 903658 663589 149130 511754 438871 > 4¹³⁸ [i]

Best Known (42, 42+91, s)-Nets in Base 4

(42, 42+91, 56)-Net over F4 — Constructive and digital

(42, 42+91, 75)-Net over F4 — Digital

(42, 42+91, 294)-Net in Base 4 — Upper bound on s

(42, 42+91, 56)-Net over F₄ — Constructive and digital

(42, 42+91, 75)-Net over F₄ — Digital