Best Known (135−86, 135, s)-Nets in Base 3
(135−86, 135, 48)-Net over F3 — Constructive and digital
Digital (49, 135, 48)-net over F3, using
- t-expansion [i] based on digital (45, 135, 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
(135−86, 135, 64)-Net over F3 — Digital
Digital (49, 135, 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
(135−86, 135, 203)-Net over F3 — Upper bound on s (digital)
There is no digital (49, 135, 204)-net over F3, because
- extracting embedded orthogonal array [i] would yield linear OA(3135, 204, F3, 86) (dual of [204, 69, 87]-code), but
- construction Y1 [i] would yield
- linear OA(3134, 166, F3, 86) (dual of [166, 32, 87]-code), but
- construction Y1 [i] would yield
- OA(3133, 150, S3, 86), but
- the linear programming bound shows that M ≥ 10206 967808 467913 747808 389212 583004 511164 705253 854573 732746 285224 754829 / 2 832691 > 3133 [i]
- OA(332, 166, S3, 16), but
- discarding factors would yield OA(332, 156, S3, 16), but
- the Rao or (dual) Hamming bound shows that M ≥ 1906 607562 901809 > 332 [i]
- discarding factors would yield OA(332, 156, S3, 16), but
- OA(3133, 150, S3, 86), but
- construction Y1 [i] would yield
- OA(369, 204, S3, 38), but
- the linear programming bound shows that M ≥ 6 710037 728131 003348 552980 099207 116544 142102 525372 997131 892145 770552 010000 / 7482 803495 454819 242924 378875 207345 317893 > 369 [i]
- linear OA(3134, 166, F3, 86) (dual of [166, 32, 87]-code), but
- construction Y1 [i] would yield
(135−86, 135, 225)-Net in Base 3 — Upper bound on s
There is no (49, 135, 226)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 26628 878185 200793 541137 584017 668580 859816 459511 287403 460269 681089 > 3135 [i]