MinT - Best Known (151−108, 151, s)-Nets in Base 4

Best Known (151−108, 151, s)-Nets in Base 4

(151−108, 151, 56)-Net over F₄ — Constructive and digital

Digital (43, 151, 56)-net over F₄, using

t-expansion [i] based on digital (33, 151, 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]

(151−108, 151, 75)-Net over F₄ — Digital

Digital (43, 151, 75)-net over F₄, using

t-expansion [i] based on digital (40, 151, 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]

(151−108, 151, 277)-Net over F₄ — Upper bound on s (digital)

There is no digital (43, 151, 278)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4¹⁵¹, 278, F₄, 108) (dual of [278, 127, 109]-code), but
- residual code [i] would yield OA(4⁴³, 169, S₄, 27), but
  - the linear programming bound shows that MÂ â‰¥ 2 258084 826110 482413 293337 625257 488476 076882 126910 259200 / 27993 955821 734889 149855 565983 > 4⁴³ [i]

(151−108, 151, 289)-Net in Base 4 — Upper bound on s

There is no (43, 151, 290)-net in base 4, because

1 times m-reduction [i] would yield (43, 150, 290)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4¹⁵⁰, 290, S₄, 107), but
  - 10 times code embedding in larger space [i] would yield OA(4¹⁶⁰, 300, S₄, 107), but
    - the linear programming bound shows that MÂ â‰¥ 3 156382 463976 497003 709343 307182 475615 499175 689524 423464 146328 443602 708192 772249 194103 952491 780634 118192 417406 939673 272036 284807 852877 919591 669822 962915 895420 208832 319747 443878 531691 781191 126259 195386 692115 582843 724901 933156 592216 282618 071242 927146 512552 886572 963526 417481 769301 010160 903627 310214 231645 979274 883913 378106 533561 881330 837407 305927 679535 335294 022404 816836 558683 655247 022119 825350 517219 452046 306079 584315 174056 498824 121895 527016 105448 093308 539445 766856 199570 468046 760630 302237 896536 790012 766366 118533 865576 980462 991827 675110 680590 200039 334257 229913 125752 581893 138754 114017 720432 022515 297932 628196 393001 480438 467045 550601 196724 227852 793573 705745 414085 398629 800178 145719 303608 815234 264369 524385 978871 582346 465600 896707 347074 831035 143730 867759 251530 381439 998834 537977 671404 904546 677069 717688 780390 692292 657152 / 1 040887 227796 701736 131935 525636 380239 022376 331794 490320 909296 014707 472932 912649 833373 083606 199985 055842 954095 733983 794839 374853 541183 196485 289555 231562 775313 852036 872126 607810 976426 409915 730786 765641 178900 200685 358517 179679 368324 885867 705123 298004 672047 342976 489158 186300 515114 135323 176528 766826 490950 836667 011033 720372 528292 580974 729938 873112 146723 807011 430713 543778 136375 206214 619320 948803 521430 538232 938848 873838 436419 262270 763837 846573 010919 322902 745185 443461 969497 941822 006894 919165 761334 594384 400206 547714 124955 325473 456912 598341 813686 995799 419355 278506 436667 738353 636251 758940 980311 340642 035778 990072 982962 564715 171868 341966 447486 642626 341461 909841 013805 428979 731325 750866 112880 725568 436359 545554 494439 740926 478125 > 4¹⁶⁰ [i]

Best Known (151−108, 151, s)-Nets in Base 4

(151−108, 151, 56)-Net over F4 — Constructive and digital

(151−108, 151, 75)-Net over F4 — Digital

(151−108, 151, 277)-Net over F4 — Upper bound on s (digital)

(151−108, 151, 289)-Net in Base 4 — Upper bound on s

(151−108, 151, 56)-Net over F₄ — Constructive and digital

(151−108, 151, 75)-Net over F₄ — Digital

(151−108, 151, 277)-Net over F₄ — Upper bound on s (digital)