MinT - Best Known (141−109, 141, s)-Nets in Base 5

Best Known (141−109, 141, s)-Nets in Base 5

(141−109, 141, 72)-Net over F₅ — Constructive and digital

Digital (32, 141, 72)-net over F₅, using

t-expansion [i] based on digital (31, 141, 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]

(141−109, 141, 276)-Net over F₅ — Upper bound on s (digital)

There is no digital (32, 141, 277)-net over F₅, because

4 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]

(141−109, 141, 290)-Net in Base 5 — Upper bound on s

There is no (32, 141, 291)-net in base 5, because

1 times m-reduction [i] would yield (32, 140, 291)-net in base 5, but
- extracting embedded orthogonal array [i] would yield OA(5¹⁴⁰, 291, S₅, 108), but
  - 9 times code embedding in larger space [i] would yield OA(5¹⁴⁹, 300, S₅, 108), but
    - the linear programming bound shows that MÂ â‰¥ 88 241721 648258 823532 919736 447346 156376 832788 611214 560208 603029 197228 807958 777684 155748 999091 978119 093105 727549 652809 836083 140458 458621 638312 952348 019440 558120 342685 949755 248955 847092 226610 421474 457719 040697 653504 065351 771521 401384 713392 416969 281573 170853 242240 179211 327880 127333 513706 512752 597396 120797 144529 427167 292202 266878 738275 804604 761264 852084 704068 989678 081654 373296 823255 076277 052228 839946 788896 130302 870512 968639 201071 158100 246395 255234 109822 499439 770274 746607 137797 035779 056867 636901 476183 087784 186191 049523 333715 006923 554455 980223 591630 778147 160563 288722 673629 894585 433486 278549 854745 511412 788261 548490 778064 924524 598883 748308 765312 746993 388706 312730 868038 199395 813654 811661 567616 424667 249746 797929 523281 594381 298634 039286 592308 972416 929678 144212 942565 604801 139159 548520 347739 177962 782088 523144 919988 745382 244059 408884 571829 765223 556712 457201 517862 302011 022643 784009 793651 528198 429651 027015 401864 438913 492291 392433 267282 689887 430103 982625 117496 506423 129350 834794 751128 399598 741833 902056 054458 896783 101071 431882 426793 365396 192898 537691 377115 593430 594465 082334 634904 288859 144921 483955 060847 750745 083371 451882 373995 411672 928395 088374 265543 945947 265606 569127 606428 991423 056375 686551 727889 882143 380590 071354 295904 259626 018281 467205 845412 793419 058061 772375 367581 844329 833984 375000 / 499432 374928 039194 152947 945591 058337 162949 344249 887152 670979 092733 069721 526730 502663 130793 780086 580708 313601 096578 700972 118131 868048 767338 462947 396363 011745 313806 163826 646203 914629 250854 280658 529663 125439 619849 395608 031880 049114 820118 268912 827047 714613 129912 125290 286061 981083 190078 796665 296740 933606 952437 204144 744479 032644 224104 944281 943521 782612 242638 101245 333749 516829 396566 152602 189491 012723 615307 910425 273643 825176 034305 365292 689833 206296 007949 903241 876992 024114 364151 722552 315420 005734 518223 183026 641766 552774 376899 387037 205981 360515 765355 052991 106300 006087 227090 602220 236712 079784 269760 904685 762138 570829 132542 868451 339389 986972 314762 331764 560302 548846 855336 450412 340849 240677 642071 185168 723941 950565 759713 519111 320198 907360 890198 207293 430627 122082 714384 537698 241240 775894 744134 065318 211984 083143 008761 142598 628455 982815 040026 291731 284686 754254 591383 787218 606777 734120 789297 566829 529718 853026 557944 399858 359195 270272 850935 464765 302055 188800 013173 106821 418135 833813 201639 781562 713368 700072 428577 269164 117183 696585 250656 978629 174742 862024 584115 951896 261046 479884 664603 156780 604805 583400 577318 917815 425357 879922 940073 199020 346690 307103 009724 786818 883181 881676 535940 973972 424739 > 5¹⁴⁹ [i]

Best Known (141−109, 141, s)-Nets in Base 5

(141−109, 141, 72)-Net over F5 — Constructive and digital

(141−109, 141, 276)-Net over F5 — Upper bound on s (digital)

(141−109, 141, 290)-Net in Base 5 — Upper bound on s

(141−109, 141, 72)-Net over F₅ — Constructive and digital

(141−109, 141, 276)-Net over F₅ — Upper bound on s (digital)