MinT - Best Known (140−81, 140, s)-Nets in Base 3

Best Known (140−81, 140, s)-Nets in Base 3

(140−81, 140, 48)-Net over F₃ — Constructive and digital

Digital (59, 140, 48)-net over F₃, using

t-expansion [i] based on digital (45, 140, 48)-net over F₃, using
- net from sequence [i] based on digital (45, 47)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
    - an explicitly constructive algebraic function field [i]

(140−81, 140, 64)-Net over F₃ — Digital

Digital (59, 140, 64)-net over F₃, using

t-expansion [i] based on digital (49, 140, 64)-net over F₃, using
- net from sequence [i] based on digital (49, 63)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 49 and N(F) ≥ 64, using
    - a curve from the manYPoints database [i]

(140−81, 140, 297)-Net in Base 3 — Upper bound on s

There is no (59, 140, 298)-net in base 3, because

extracting embedded orthogonal array [i] would yield OA(3¹⁴⁰, 298, S₃, 81), but
- 2 times code embedding in larger space [i] would yield OA(3¹⁴², 300, S₃, 81), but
  - the linear programming bound shows that MÂ â‰¥ 1036 483355 493171 432468 543778 798304 812962 901004 506800 316185 700395 635910 832645 818420 093255 678003 628049 693231 951479 388202 025869 855064 221220 245100 031454 214448 929548 748283 874370 476918 207471 123067 768245 704454 036583 064728 995480 754088 596517 369763 924922 395932 254428 331560 645929 199108 351498 175123 266033 232675 198740 048603 127537 038044 278149 775429 986838 962322 542708 953068 256022 395660 844601 946965 352598 276041 105592 906213 316756 400250 247304 178668 634225 427732 030749 051728 050199 420099 167742 506952 613778 770599 634687 841069 013099 923860 333130 605470 420285 981504 658660 873476 825465 516752 301660 833995 611561 973974 468243 847972 242828 470632 006760 799737 588158 523861 978462 678369 525074 636292 608533 234458 458507 012558 976988 801044 988767 118480 065372 440703 913819 087954 211407 367051 178278 466779 398284 410464 750102 116704 721668 653810 839140 570943 255470 827666 005261 507501 183202 703386 433440 958508 228530 136997 243668 194549 106981 747996 962518 414673 070827 700845 663383 515426 113344 913413 987132 847588 716621 876922 867059 613569 731029 454893 708033 204116 708377 361342 668366 743124 567040 454851 779516 943193 148428 070421 701899 276785 652822 374272 168128 477700 635039 357683 998515 744085 063864 375039 425661 624994 757133 611475 428415 047999 572199 393216 370360 439368 932017 341289 823410 106650 949697 648780 483040 987151 338856 008570 963543 896406 252801 404080 073850 118445 360655 033768 634956 752276 562931 960154 285091 487417 185504 502240 / 10 795572 257332 967630 606827 060274 694971 907850 137542 554500 779290 615960 256951 061156 430348 056300 342581 383301 789407 810394 254957 766839 301395 471147 676064 149852 041015 250427 830336 090585 489247 911447 634013 446797 101073 329857 366861 891243 755261 092754 007802 242599 198542 000223 225676 823526 571704 368907 281190 035711 236023 656609 748556 152305 361198 799956 678735 982390 021925 413758 305415 759085 249523 072633 606234 875311 509275 954229 016778 531491 035672 018180 722793 693290 034988 524248 215253 126724 943906 369163 601395 210121 280193 120225 032017 442014 176930 220121 479201 341452 458464 374844 746152 583674 672984 526922 821058 577075 122839 447845 473340 432946 366223 407585 619614 738892 117112 449441 773911 134167 208125 406938 449215 570076 709829 869942 124829 866903 226335 263775 759494 002014 414341 926366 996493 858258 783626 124938 969019 091817 626055 978646 873144 981956 440744 939264 769618 134893 149711 371749 825962 039270 229764 329220 354388 470630 477613 011318 425862 335845 183990 741885 829056 464862 050731 897843 753622 031247 091657 459254 843597 622598 815761 473931 861502 011241 728895 656177 990945 179789 468325 309153 327148 248898 539318 983449 737069 061086 890758 106992 676478 271759 590547 683965 672071 826576 418629 187791 753843 430897 885335 701548 430556 789936 835446 246086 624053 745497 756689 974998 486196 444034 914305 032840 022496 162021 840569 736998 610154 149565 252702 138825 240111 > 3¹⁴² [i]

Best Known (140−81, 140, s)-Nets in Base 3

(140−81, 140, 48)-Net over F3 — Constructive and digital

(140−81, 140, 64)-Net over F3 — Digital

(140−81, 140, 297)-Net in Base 3 — Upper bound on s

(140−81, 140, 48)-Net over F₃ — Constructive and digital

(140−81, 140, 64)-Net over F₃ — Digital