MinT - Best Known (82, 82+∞, s)-Nets in Base 9

Best Known (82, 82+∞, s)-Nets in Base 9

(82, 82+∞, 222)-Net over F₉ — Constructive and digital

Digital (82, m, 222)-net over F₉ for arbitrarily large m, using

net from sequence [i] based on digital (82, 221)-sequence over F₉, using
- t-expansion [i] based on digital (79, 221)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 79 and N(F) ≥ 222, using
    - F₄ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(82, 82+∞, 245)-Net over F₉ — Digital

Digital (82, m, 245)-net over F₉ for arbitrarily large m, using

net from sequence [i] based on digital (82, 244)-sequence over F₉, using
- t-expansion [i] based on digital (81, 244)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 81 and N(F) ≥ 245, using
    - Drinfeld modules of rank 1 [i]

(82, 82+∞, 683)-Net in Base 9 — Upper bound on s

There is no (82, m, 684)-net in base 9 for arbitrarily large m, because

m-reduction [i] would yield (82, 2048, 684)-net in base 9, but
- extracting embedded OOA [i] would yield OOA(9²⁰⁴⁸, 684, S₉, 3, 1966), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 48 809689 446507 161719 089886 203712 925406 523333 009542 346913 873757 029357 602182 640564 022200 214888 757614 644732 059092 438828 432232 284218 117248 854706 824287 790045 625641 478932 646777 963537 266319 636464 756139 597149 390311 613921 024590 382134 146154 403459 058037 974155 122177 608550 816945 369588 589724 015669 782030 973619 577072 168183 019745 542266 152815 070314 803748 269351 473741 766645 348709 971015 305229 713564 979151 851003 939293 771586 963853 316433 514854 971694 978447 440728 061267 976694 254308 174891 560012 248090 321999 773067 730734 285628 514771 090404 746162 463627 388039 849327 419896 605183 619895 059408 062685 182734 896881 169533 355913 164487 802993 086035 119025 610428 666798 351217 188412 963222 383667 206913 576355 353764 916582 415443 726285 299065 421594 608206 924346 038523 558250 624145 055099 759736 596830 243162 794009 010615 331219 969064 871290 478311 171786 699449 825940 785364 955778 786478 835716 352430 157207 035556 681989 617222 830134 910205 356331 543484 026317 765760 045965 019931 572586 854520 718619 615563 140386 559361 811240 322215 604112 038997 691631 755217 746655 764045 869647 517352 019754 038046 490205 326888 114636 330957 789210 009675 468141 804573 629654 087113 326752 450975 196424 400879 196970 811281 621877 226441 418515 496706 564228 657328 309847 903803 727909 566968 873802 823087 892145 467247 668836 617906 899625 655213 855833 611702 002911 398721 211344 742761 335056 995880 492701 906291 148722 406809 714545 476259 022979 915648 002161 219853 568785 551137 554368 806680 714696 062942 446442 742377 876853 606534 475019 991172 300551 500668 662872 338667 338469 393625 646760 858909 434429 547819 843822 318300 128336 345178 528027 788176 886507 753569 572308 954282 900164 969366 826769 521339 766994 482762 229647 247592 873594 288129 038804 514163 087321 930807 704010 032789 899667 622390 770417 678962 370790 225180 940801 361897 477719 446847 554829 316086 904977 821123 558148 136773 956916 808717 967613 734597 033904 749127 735100 901656 136781 217197 944287 045255 709693 252336 810501 965636 838546 171905 512581 767640 266266 085297 853392 387605 051502 848275 418253 053003 894811 327359 438274 395815 859949 562521 802650 150868 964641 552605 635546 361940 462031 / 1967 > 9²⁰⁴⁸ [i]

Best Known (82, 82+∞, s)-Nets in Base 9

(82, 82+∞, 222)-Net over F9 — Constructive and digital

(82, 82+∞, 245)-Net over F9 — Digital

(82, 82+∞, 683)-Net in Base 9 — Upper bound on s

(82, 82+∞, 222)-Net over F₉ — Constructive and digital

(82, 82+∞, 245)-Net over F₉ — Digital