MinT - Best Known (83, ∞, s)-Nets in Base 8

Best Known (83, ∞, s)-Nets in Base 8

(83, ∞, 98)-Net over F₈ — Constructive and digital

Digital (83, m, 98)-net over F₈ for arbitrarily large m, using

net from sequence [i] based on digital (83, 97)-sequence over F₈, using
- t-expansion [i] based on digital (37, 97)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 37 and N(F) ≥ 98, using
    - F₅ from the tower of function fields over F₈ by van der Geer and van der Vlugt [i]

(83, ∞, 104)-Net in Base 8 — Constructive

(83, m, 104)-net in base 8 for arbitrarily large m, using

net from sequence [i] based on (83, 103)-sequence in base 8, using
- base expansion [i] based on digital (249, 103)-sequence over F₂, using
  - base reduction for sequences [i] based on digital (73, 103)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
      - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(83, ∞, 195)-Net over F₈ — Digital

Digital (83, m, 195)-net over F₈ for arbitrarily large m, using

net from sequence [i] based on digital (83, 194)-sequence over F₈, using
- t-expansion [i] based on digital (77, 194)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 77 and N(F) ≥ 195, using
    - Drinfeld modules of rank 1 [i]

(83, ∞, 606)-Net in Base 8 — Upper bound on s

There is no (83, m, 607)-net in base 8 for arbitrarily large m, because

m-reduction [i] would yield (83, 1817, 607)-net in base 8, but
- extracting embedded OOA [i] would yield OOA(8¹⁸¹⁷, 607, S₈, 3, 1734), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 1 833160 769754 882874 011377 193031 793086 719480 374186 535856 831995 282342 427703 569960 888193 613818 547447 642758 136190 352087 575658 594382 807049 770068 332149 331738 605416 807322 492673 284798 787928 325250 149641 050647 925927 339237 338492 263522 681352 179848 586775 846681 496751 248484 428450 328617 247664 869939 625564 068065 907106 488407 348385 540178 649663 430332 132802 419007 465161 445154 032277 411445 741960 909272 918936 260134 825993 612771 827025 862492 728123 108531 816159 218008 239730 391798 046717 317308 859833 300219 050238 471026 004994 943401 854054 669970 637299 674966 513353 074031 787113 116622 144404 635463 218450 285540 643404 096034 722923 602653 716091 525957 626090 422342 041964 412598 390957 234741 038071 203572 542455 719855 737446 906676 646196 784922 930982 925827 706637 906943 722077 249659 859994 662473 131406 870964 471303 973941 166836 742195 561546 797644 330707 707863 267005 727941 228089 666412 216637 383120 954634 721625 431327 320771 412217 563656 902480 705744 514560 275117 910991 136764 712958 061116 492171 333958 307940 422850 499709 783086 802183 744706 265038 142548 486812 128039 958392 847013 724024 439149 880701 973182 746088 323327 679156 508068 489122 944585 060987 537055 950786 804365 407395 241358 318772 117622 292446 182450 149343 259867 068888 379588 184171 593737 215691 305157 273405 498933 389967 833725 725016 608866 151476 807611 774651 769852 334167 837448 732149 731989 406396 909257 122238 737262 428688 326743 257560 183673 693348 480881 114590 342843 078633 394492 545452 978498 745728 363138 062447 186749 143626 097083 832423 273899 790530 525976 739297 846952 873232 318834 160536 559011 862534 206580 309734 569290 015485 452236 537290 983238 212887 794024 638604 580621 107142 582598 544629 154520 308392 875262 085227 888435 952822 435021 896905 838497 138509 856336 538692 254151 707505 147794 068816 083116 097536 / 1735 > 8¹⁸¹⁷ [i]

Best Known (83, ∞, s)-Nets in Base 8

(83, ∞, 98)-Net over F8 — Constructive and digital

(83, ∞, 104)-Net in Base 8 — Constructive

(83, ∞, 195)-Net over F8 — Digital

(83, ∞, 606)-Net in Base 8 — Upper bound on s

(83, ∞, 98)-Net over F₈ — Constructive and digital

(83, ∞, 195)-Net over F₈ — Digital