MinT - Best Known (109−94, 109, s)-Nets in Base 32

Best Known (109−94, 109, s)-Nets in Base 32

(109−94, 109, 120)-Net over F₃₂ — Constructive and digital

Digital (15, 109, 120)-net over F₃₂, using

t-expansion [i] based on digital (11, 109, 120)-net over F₃₂, using
- net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
    - a function field by SÃ©mirat [i]

(109−94, 109, 158)-Net over F₃₂ — Digital

Digital (15, 109, 158)-net over F₃₂, using

net from sequence [i] based on digital (15, 157)-sequence over F₃₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 15 and N(F) ≥ 158, using
  - an algebraic function field by Beelen and Pellikaan [i]

(109−94, 109, 1804)-Net over F₃₂ — Upper bound on s (digital)

There is no digital (15, 109, 1805)-net over F₃₂, because

extracting embedded orthogonal array [i] would yield linear OA(32¹⁰⁹, 1805, F₃₂, 94) (dual of [1805, 1696, 95]-code), but
- the Johnson bound shows that NÂ â‰¤ 538 787694 062958 461577 254464 453834 300387 090815 644475 674058 325408 323591 107858 098529 994849 273390 958626 388000 282595 851337 268743 839149 824310 870769 503449 734404 851389 366344 325645 578100 029714 793319 195806 916571 846144 980816 935367 065089 205228 292034 440956 038090 731747 203150 593066 774730 005599 444896 306179 447680 242877 369111 546409 961557 101683 551636 792327 577657 148554 449303 338643 471284 648473 912362 079767 826048 722301 979361 368347 621712 402859 142559 432586 597992 274319 038067 087716 564700 002718 484088 781773 301566 662830 916748 127673 568889 052929 851759 129869 902525 186175 053943 250988 273058 903189 856152 779945 623427 035108 764931 802616 408233 486327 232647 354742 715263 274818 363260 012142 457780 773974 956742 883157 380728 396958 451074 491961 874225 976596 227161 563091 604246 078475 950828 904021 654309 440501 129040 669542 360630 770906 166045 577799 982761 088507 319459 257212 179868 175729 259136 680450 895611 877340 856740 156112 119846 427870 573960 215074 383582 576915 977785 562573 829933 812900 016813 156643 690108 160018 509114 108845 969376 503314 262404 528737 398372 429196 606471 814239 556344 134027 948289 123670 057164 567402 966505 839048 482430 469852 578460 899532 386175 457500 336925 768125 572905 477686 021935 179308 942047 623502 066454 132267 103472 125578 303249 292658 297096 427280 303429 738567 407376 749755 942805 173442 419480 424998 870286 211441 660040 779171 924416 095466 152916 816529 517913 396461 797212 709871 311500 709237 994406 556504 355342 285854 398206 104636 176751 739080 007852 576131 906229 584581 257751 780443 331405 806835 999465 335879 527724 957994 776570 372284 384468 372124 647568 877676 923229 181453 130876 874398 083599 415119 879347 104448 938514 743391 702064 219484 053905 868452 482921 125713 928832 721051 655793 260115 971142 460184 207396 618426 212885 018392 712939 472592 619468 189523 958280 620472 473103 825488 893190 970423 474659 084504 905852 406776 174028 804926 873602 727109 467386 746618 964235 635539 482623 324128 222569 301772 270251 951892 989493 156492 223202 516139 286278 022765 558797 703252 353319 874843 854969 639267 294366 978196 009315 054784 184984 799085 615972 232341 938038 066502 510017 552033 208122 495427 317672 935724 165060 136747 520955 502074 091520 221141 439624 636121 552976 424715 464110 326771 932068 398930 859540 837207 975437 522703 117301 956573 804323 564820 039778 658504 029986 714184 438761 467776 268891 757386 877278 997205 201263 544012 849203 535409 925766 958188 553920 860344 947590 027672 192795 867494 070791 765252 024379 921010 189114 961181 098620 362299 005677 144110 137942 214166 180059 833085 989615 634491 183920 199825 414678 321039 658294 096851 792476 526247 140222 375902 358602 996812 638267 066746 243249 728132 198186 147663 392153 426168 197700 521436 238825 596182 327474 793615 423803 047778 982969 859385 785681 431817 885280 285730 794986 700208 903826 < 32¹⁶⁹⁶ [i]

(109−94, 109, 1809)-Net in Base 32 — Upper bound on s

There is no (15, 109, 1810)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 116 431287 458834 561643 192094 357564 400495 620832 774888 621050 335180 687210 195101 210215 687700 038450 731655 031113 731704 320686 556211 628896 255392 921989 427370 505428 141866 022592 > 32¹⁰⁹ [i]

Best Known (109−94, 109, s)-Nets in Base 32

(109−94, 109, 120)-Net over F32 — Constructive and digital

(109−94, 109, 158)-Net over F32 — Digital

(109−94, 109, 1804)-Net over F32 — Upper bound on s (digital)

(109−94, 109, 1809)-Net in Base 32 — Upper bound on s

(109−94, 109, 120)-Net over F₃₂ — Constructive and digital

(109−94, 109, 158)-Net over F₃₂ — Digital

(109−94, 109, 1804)-Net over F₃₂ — Upper bound on s (digital)