Best Known (181−118, 181, s)-Nets in Base 3
(181−118, 181, 48)-Net over F3 — Constructive and digital
Digital (63, 181, 48)-net over F3, using
- t-expansion [i] based on digital (45, 181, 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
(181−118, 181, 64)-Net over F3 — Digital
Digital (63, 181, 64)-net over F3, using
- t-expansion [i] based on digital (49, 181, 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
(181−118, 181, 227)-Net over F3 — Upper bound on s (digital)
There is no digital (63, 181, 228)-net over F3, because
- 1 times m-reduction [i] would yield digital (63, 180, 228)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3180, 228, F3, 117) (dual of [228, 48, 118]-code), but
- residual code [i] would yield OA(363, 110, S3, 39), but
- the linear programming bound shows that M ≥ 401 475818 140604 282392 040969 710232 145438 658389 807791 365215 582135 475407 151345 717325 294327 153062 902549 711657 673773 152251 547301 / 325 455230 806957 608121 722211 436046 187202 554897 940196 290012 728886 526860 582095 622224 001861 980520 > 363 [i]
- residual code [i] would yield OA(363, 110, S3, 39), but
- extracting embedded orthogonal array [i] would yield linear OA(3180, 228, F3, 117) (dual of [228, 48, 118]-code), but
(181−118, 181, 277)-Net in Base 3 — Upper bound on s
There is no (63, 181, 278)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 257 512190 137533 930002 023002 674502 556875 876375 953218 517665 440369 655595 130250 263633 676017 > 3181 [i]