Best Known (87−77, 87, s)-Nets in Base 27
(87−77, 87, 94)-Net over F27 — Constructive and digital
Digital (10, 87, 94)-net over F27, using
- net from sequence [i] based on digital (10, 93)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 10 and N(F) ≥ 94, using
(87−77, 87, 99)-Net over F27 — Digital
Digital (10, 87, 99)-net over F27, using
- t-expansion [i] based on digital (9, 87, 99)-net over F27, using
- net from sequence [i] based on digital (9, 98)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 9 and N(F) ≥ 99, using
- net from sequence [i] based on digital (9, 98)-sequence over F27, using
(87−77, 87, 978)-Net over F27 — Upper bound on s (digital)
There is no digital (10, 87, 979)-net over F27, because
- 5 times m-reduction [i] would yield digital (10, 82, 979)-net over F27, but
- extracting embedded orthogonal array [i] would yield linear OA(2782, 979, F27, 72) (dual of [979, 897, 73]-code), but
- the Johnson bound shows that N ≤ 841606 457614 836711 867150 367477 729477 401026 596406 497135 630376 023737 879437 461728 919684 884705 500268 662379 337865 811116 718919 454593 850158 047721 116667 379238 614250 251875 789484 335398 495916 774404 425236 982494 546179 521927 468463 531736 116694 388054 585099 830995 316234 002691 771138 164533 672810 017181 985549 118157 728661 773759 793530 357895 783971 875164 401451 732385 838070 864732 983492 193483 076983 655522 086909 142581 249272 770118 165466 146725 426608 202382 933056 066267 940167 876118 436024 397971 092045 945252 212869 582793 292630 241984 297503 488064 637506 324188 518582 296963 980183 179965 268008 368235 997773 794328 232347 675802 267134 011284 474679 407757 507696 509151 249158 023382 709054 895225 758748 811156 227810 208946 358691 671889 337588 241477 443380 589749 262699 546981 844574 774426 072955 980109 272344 970755 985235 923878 356827 141692 243721 791642 554372 175716 975091 242405 997261 120864 932276 119932 436914 000368 010752 005644 990616 580061 482309 158642 182491 159561 370702 410368 406327 685972 115768 194894 633408 418347 372182 547081 735397 425544 946867 965913 326580 759733 527142 417203 566814 740136 236588 590929 341900 727134 898117 249710 587284 514007 042638 677916 653759 964465 191088 533180 633231 734137 422161 817292 228562 373033 737573 718097 763999 587240 740391 332742 557859 765700 639142 602640 312389 382640 712887 665812 271720 588444 155790 271115 355223 692178 749278 747427 104262 727062 201019 < 27897 [i]
- extracting embedded orthogonal array [i] would yield linear OA(2782, 979, F27, 72) (dual of [979, 897, 73]-code), but
(87−77, 87, 981)-Net in Base 27 — Upper bound on s
There is no (10, 87, 982)-net in base 27, because
- 3 times m-reduction [i] would yield (10, 84, 982)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 1 725017 694307 848830 536857 223602 738159 718846 790703 668433 852685 580157 674163 604630 375884 986530 803870 173418 211846 835945 871189 > 2784 [i]