Best Known (59, 59+108, s)-Nets in Base 3
(59, 59+108, 48)-Net over F3 — Constructive and digital
Digital (59, 167, 48)-net over F3, using
- t-expansion [i] based on digital (45, 167, 48)-net over F3, using
- net from sequence [i] based on digital (45, 47)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 45 and N(F) ≥ 48, using
- net from sequence [i] based on digital (45, 47)-sequence over F3, using
(59, 59+108, 64)-Net over F3 — Digital
Digital (59, 167, 64)-net over F3, using
- t-expansion [i] based on digital (49, 167, 64)-net over F3, using
- net from sequence [i] based on digital (49, 63)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 49 and N(F) ≥ 64, using
- net from sequence [i] based on digital (49, 63)-sequence over F3, using
(59, 59+108, 225)-Net over F3 — Upper bound on s (digital)
There is no digital (59, 167, 226)-net over F3, because
- extracting embedded orthogonal array [i] would yield linear OA(3167, 226, F3, 108) (dual of [226, 59, 109]-code), but
- residual code [i] would yield OA(359, 117, S3, 36), but
- the linear programming bound shows that M ≥ 761 727308 447698 482542 148743 230569 434155 182882 786455 076206 353848 581403 820807 661963 484751 862555 171263 099957 412829 349685 769072 084648 120157 630433 734361 323683 270259 221443 291357 973630 261156 160974 216627 852781 876417 362687 833375 760246 069924 / 51122 849719 374140 322887 143047 504511 568594 316413 366999 782333 439013 284638 072569 059057 230504 528941 253115 325619 904540 874976 816730 143273 988924 400609 746921 908790 318590 477735 687435 119706 723653 511140 691154 502149 > 359 [i]
- residual code [i] would yield OA(359, 117, S3, 36), but
(59, 59+108, 263)-Net in Base 3 — Upper bound on s
There is no (59, 167, 264)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 53 450920 021509 008789 933715 360116 914812 576964 449229 137946 655143 413518 165487 335217 > 3167 [i]