Best Known (81−72, 81, s)-Nets in Base 27
(81−72, 81, 88)-Net over F27 — Constructive and digital
Digital (9, 81, 88)-net over F27, using
- net from sequence [i] based on digital (9, 87)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 9 and N(F) ≥ 88, using
(81−72, 81, 99)-Net over F27 — Digital
Digital (9, 81, 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
(81−72, 81, 887)-Net over F27 — Upper bound on s (digital)
There is no digital (9, 81, 888)-net over F27, because
- 4 times m-reduction [i] would yield digital (9, 77, 888)-net over F27, but
- extracting embedded orthogonal array [i] would yield linear OA(2777, 888, F27, 68) (dual of [888, 811, 69]-code), but
- the Johnson bound shows that N ≤ 676 073494 111179 155944 831344 320678 285290 282742 320342 427959 865433 047311 864058 108998 120018 864964 271413 607203 429856 319710 106304 050723 790165 435618 723517 505266 847536 006734 019758 191203 370927 317924 151147 146873 669953 329742 210813 862587 763640 823380 213628 414999 445124 815704 706320 732721 489984 430386 626654 729804 682503 317287 705776 706882 782666 304852 283523 977648 670659 107013 718867 683031 147195 482998 355791 949127 742911 073258 395174 512269 986144 543716 856181 042140 661426 711094 771210 514367 063617 358330 218320 990585 294038 147506 416705 224519 015719 543865 558951 378390 831481 745897 054929 089584 567280 537600 055782 106918 661650 902595 992951 361196 271021 410706 833165 020373 919776 425861 224339 751607 392542 705275 673715 779740 322262 844804 229959 552755 028380 164626 494385 723436 131752 919866 042187 250140 836017 834060 706825 916117 327972 196218 257564 975303 263068 486761 775445 278150 507787 745015 695260 327247 092874 118713 690403 613851 454225 864275 348691 290604 626499 565053 191864 842395 385547 138595 718994 337948 112415 955301 424370 616043 177533 823467 491972 338448 662335 941935 501858 942041 405321 840843 045608 844744 225514 880640 367429 270344 213292 863991 899622 158874 334767 298051 666030 146395 102355 149116 062819 093515 507264 745338 948039 275454 < 27811 [i]
- extracting embedded orthogonal array [i] would yield linear OA(2777, 888, F27, 68) (dual of [888, 811, 69]-code), but
(81−72, 81, 889)-Net in Base 27 — Upper bound on s
There is no (9, 81, 890)-net in base 27, because
- 6 times m-reduction [i] would yield (9, 75, 890)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 228863 545622 381993 226669 535478 212825 885166 201251 437077 552420 519466 876310 969746 511362 429346 749504 006207 664933 > 2775 [i]