Best Known (18, 18+103, s)-Nets in Base 8
(18, 18+103, 65)-Net over F8 — Constructive and digital
Digital (18, 121, 65)-net over F8, using
- t-expansion [i] based on digital (14, 121, 65)-net over F8, using
- net from sequence [i] based on digital (14, 64)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 14 and N(F) ≥ 65, using
- the Suzuki function field over F8 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 14 and N(F) ≥ 65, using
- net from sequence [i] based on digital (14, 64)-sequence over F8, using
(18, 18+103, 235)-Net over F8 — Upper bound on s (digital)
There is no digital (18, 121, 236)-net over F8, because
- extracting embedded orthogonal array [i] would yield linear OA(8121, 236, F8, 103) (dual of [236, 115, 104]-code), but
- construction Y1 [i] would yield
- OA(8120, 138, S8, 103), but
- the linear programming bound shows that M ≥ 111645 453241 746255 920862 684222 479803 153865 164146 020446 912158 451311 961713 255226 562612 641032 127218 234498 669820 214273 467205 287936 / 44751 440946 265625 > 8120 [i]
- OA(8115, 236, S8, 98), but
- discarding factors would yield OA(8115, 147, S8, 98), but
- the linear programming bound shows that M ≥ 599345 416299 388973 235507 022177 177118 046850 911577 140306 689015 460811 889080 427692 784047 046445 821063 607212 795117 279723 192210 808892 966020 579328 / 7743 507384 508665 125912 260586 578125 > 8115 [i]
- discarding factors would yield OA(8115, 147, S8, 98), but
- OA(8120, 138, S8, 103), but
- construction Y1 [i] would yield
(18, 18+103, 295)-Net in Base 8 — Upper bound on s
There is no (18, 121, 296)-net in base 8, because
- 5 times m-reduction [i] would yield (18, 116, 296)-net in base 8, but
- extracting embedded orthogonal array [i] would yield OA(8116, 296, S8, 98), but
- 4 times code embedding in larger space [i] would yield OA(8120, 300, S8, 98), but
- the linear programming bound shows that M ≥ 72 391436 036661 870506 841663 284749 383948 996578 497360 045511 574810 433100 255329 333527 458922 411946 454865 643457 540180 726661 803342 026848 614599 345418 648840 804419 427140 608273 178357 015888 950266 136667 655834 011931 483579 565276 837676 109005 285853 390154 500910 063042 730800 034201 395522 890927 060349 838070 912163 028243 147934 409475 727494 923833 324123 955969 079425 802665 504019 003172 537304 266547 419992 973549 685854 570582 518651 067108 338295 135301 669741 318242 472033 455161 862112 610613 107184 093100 572061 540997 398528 / 29 833571 454284 666730 273810 076668 595022 381000 861479 193643 318605 828254 959929 285703 016807 839251 126709 255146 942998 619648 119868 620866 035186 717895 239752 226492 992926 837847 710783 960610 903347 533279 891708 855922 743885 653623 024410 992653 044103 197695 890752 666761 781784 319919 703649 673999 906001 435357 616750 918757 442488 337081 208663 397951 043953 064284 007145 394137 438180 886383 843011 053613 936443 > 8120 [i]
- 4 times code embedding in larger space [i] would yield OA(8120, 300, S8, 98), but
- extracting embedded orthogonal array [i] would yield OA(8116, 296, S8, 98), but