Best Known (75−66, 75, s)-Nets in Base 27
(75−66, 75, 88)-Net over F27 — Constructive and digital
Digital (9, 75, 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
(75−66, 75, 99)-Net over F27 — Digital
Digital (9, 75, 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
(75−66, 75, 888)-Net over F27 — Upper bound on s (digital)
There is no digital (9, 75, 889)-net over F27, because
- extracting embedded orthogonal array [i] would yield linear OA(2775, 889, F27, 66) (dual of [889, 814, 67]-code), but
- the Johnson bound shows that N ≤ 13 272118 574412 907097 684970 795776 881237 037353 172326 850229 637005 130675 014606 247892 455783 845339 131928 829679 431555 942966 487232 245416 506293 053419 438051 006214 396818 449096 658968 200057 727831 698899 184751 770237 162109 223432 527104 675120 706665 917645 600168 246299 215746 271705 252047 483628 553083 305027 286417 752352 144155 467001 691428 146882 048699 500830 630763 644971 931732 930201 533888 405895 701628 589929 322645 244693 781637 148125 350939 467265 562130 424261 388214 541392 423509 783132 636102 958466 127581 018549 888997 200126 683730 695311 718177 653453 137113 561882 532968 849743 766199 595015 542418 318713 691538 182000 497071 767270 368402 160166 488677 562427 854316 263861 842481 253901 093615 474968 705654 921102 574160 716727 137046 329862 171575 886122 454329 796860 679217 675333 345280 236390 330860 985890 285820 941254 136126 684995 297851 447282 007481 019198 002815 164602 103786 770485 970425 802257 798056 227555 561785 470235 042318 737376 606438 617002 448069 825388 433157 400519 738972 900106 374542 570534 750150 761806 222564 749177 510298 332133 006589 046897 632669 946842 630335 060462 198472 292418 281118 244265 581897 873024 575003 705809 002856 516693 837215 000622 313225 762218 838403 484529 910930 605672 437195 793691 232248 581487 295704 665281 382074 963297 442670 435471 978602 < 27814 [i]
(75−66, 75, 889)-Net in Base 27 — Upper bound on s
There is no (9, 75, 890)-net in base 27, because
- 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]