Best Known (54, 54+98, s)-Nets in Base 3
(54, 54+98, 48)-Net over F3 — Constructive and digital
Digital (54, 152, 48)-net over F3, using
- t-expansion [i] based on digital (45, 152, 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
(54, 54+98, 64)-Net over F3 — Digital
Digital (54, 152, 64)-net over F3, using
- t-expansion [i] based on digital (49, 152, 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
(54, 54+98, 222)-Net over F3 — Upper bound on s (digital)
There is no digital (54, 152, 223)-net over F3, because
- 2 times m-reduction [i] would yield digital (54, 150, 223)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3150, 223, F3, 96) (dual of [223, 73, 97]-code), but
- residual code [i] would yield OA(354, 126, S3, 32), but
- the linear programming bound shows that M ≥ 15 551175 343489 816312 839781 958526 754046 650025 716899 066271 597025 177434 186311 225436 494080 388171 855211 042571 528095 306878 344834 964322 752687 664073 776320 / 247669 228492 701837 800287 038860 817164 439353 182480 938178 042680 577677 782903 404567 720365 536303 663330 806259 025721 550363 325847 > 354 [i]
- residual code [i] would yield OA(354, 126, S3, 32), but
- extracting embedded orthogonal array [i] would yield linear OA(3150, 223, F3, 96) (dual of [223, 73, 97]-code), but
(54, 54+98, 243)-Net in Base 3 — Upper bound on s
There is no (54, 152, 244)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 3 794172 348114 038262 098082 438932 542068 702242 879080 975870 695235 237120 777449 > 3152 [i]