MinT - Best Known (139−107, 139, s)-Nets in Base 5

Best Known (139−107, 139, s)-Nets in Base 5

(139−107, 139, 72)-Net over F₅ — Constructive and digital

Digital (32, 139, 72)-net over F₅, using

t-expansion [i] based on digital (31, 139, 72)-net over F₅, using
- net from sequence [i] based on digital (31, 71)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 31 and N(F) ≥ 72, using
    - an explicitly constructive algebraic function field [i]

(139−107, 139, 276)-Net over F₅ — Upper bound on s (digital)

There is no digital (32, 139, 277)-net over F₅, because

2 times m-reduction [i] would yield digital (32, 137, 277)-net over F₅, but
- extracting embedded orthogonal array [i] would yield linear OA(5¹³⁷, 277, F₅, 105) (dual of [277, 140, 106]-code), but
  - residual code [i] would yield OA(5³², 171, S₅, 21), but
    - 1 times truncation [i] would yield OA(5³¹, 170, S₅, 20), but
      - the linear programming bound shows that MÂ â‰¥ 42216 069885 091800 891878 320552 825927 734375 / 8 765861 509813 289221 > 5³¹ [i]

(139−107, 139, 291)-Net in Base 5 — Upper bound on s

There is no (32, 139, 292)-net in base 5, because

2 times m-reduction [i] would yield (32, 137, 292)-net in base 5, but
- extracting embedded orthogonal array [i] would yield OA(5¹³⁷, 292, S₅, 105), but
  - 8 times code embedding in larger space [i] would yield OA(5¹⁴⁵, 300, S₅, 105), but
    - the linear programming bound shows that MÂ â‰¥ 290 968901 921218 585235 010533 763546 870874 254352 170153 535490 757597 079148 423892 679339 207218 266512 700960 563001 774847 403846 954854 962301 073274 097109 652566 671532 140140 288569 789558 185687 774209 179349 957062 674845 820680 603233 802447 061784 495828 188406 601821 618699 761185 551434 687982 805240 076439 551679 167511 383360 409325 989172 274720 757318 880441 749280 924785 217100 497426 607808 032938 508597 582657 515623 941302 551952 664175 267073 527452 206472 860127 621569 890215 940981 822501 759293 740545 253373 957766 225373 396610 565602 738987 015343 968196 745009 546183 948997 519789 973505 776239 648003 891536 198362 353598 951983 120720 999144 496434 337996 123780 307820 663657 860311 344200 803692 248558 610424 982937 748829 237969 556414 142573 277968 322022 110325 035879 056108 551824 438830 442720 483316 536101 775075 645420 510240 082554 916183 953957 289331 674952 227966 041731 551714 556454 347602 125029 685836 068037 287763 076856 631815 284402 202942 064262 490453 163332 957934 172037 419074 202213 466919 416379 317718 766010 626490 187144 321258 500311 280787 140416 553784 171338 467191 978320 689720 014577 646256 174183 835802 871941 970204 414927 239142 429008 225093 051801 629307 287556 637126 279470 644762 573048 736537 442079 188050 345258 055883 815404 075677 638708 772901 083616 191294 557335 007202 906587 832904 952602 872167 123182 305104 034384 157785 098068 416118 621826 171875 / 829 407844 108514 116566 912543 870634 011524 497112 448587 785813 743871 640196 436361 297654 842291 165011 829090 178389 368593 997340 169565 791261 319098 716781 763873 391000 796439 364885 110745 747722 673179 951936 163734 282454 534859 368378 869843 797447 811638 984269 949649 948585 400334 182284 457283 390329 967601 776279 857630 316543 044214 037087 207431 434563 371538 559338 094814 077272 326971 437196 498814 675515 361638 039058 742227 263032 407672 167769 349431 669669 752951 842601 528227 950372 434195 860470 265189 256312 894315 797257 995006 045418 497433 111340 528431 212242 542326 343030 304369 737789 091920 770439 814959 803515 033243 559521 203463 706691 923202 489559 836647 921985 917038 528685 343961 953115 448831 302820 256143 197423 630060 635015 344196 311662 206231 943038 073402 541609 470646 711833 436188 108223 673785 350130 771991 214249 926941 416646 933145 768815 254834 060357 116221 794511 836463 304356 906138 197138 030965 557937 166197 085295 653229 826858 453386 256404 851806 496956 448404 231373 128448 715682 488374 750119 667979 356825 955862 317668 855900 017837 325091 445289 723294 371775 002912 220689 474835 912739 678174 388543 926883 429416 832931 017366 171049 477953 540926 284917 202169 783234 697130 915924 224231 148459 956876 641130 040143 970750 705067 574552 950549 > 5¹⁴⁵ [i]

Best Known (139−107, 139, s)-Nets in Base 5

(139−107, 139, 72)-Net over F5 — Constructive and digital

(139−107, 139, 276)-Net over F5 — Upper bound on s (digital)

(139−107, 139, 291)-Net in Base 5 — Upper bound on s

(139−107, 139, 72)-Net over F₅ — Constructive and digital

(139−107, 139, 276)-Net over F₅ — Upper bound on s (digital)