Best Known (56, 56+106, s)-Nets in Base 3
(56, 56+106, 48)-Net over F3 — Constructive and digital
Digital (56, 162, 48)-net over F3, using
- t-expansion [i] based on digital (45, 162, 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
(56, 56+106, 64)-Net over F3 — Digital
Digital (56, 162, 64)-net over F3, using
- t-expansion [i] based on digital (49, 162, 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
(56, 56+106, 203)-Net over F3 — Upper bound on s (digital)
There is no digital (56, 162, 204)-net over F3, because
- 1 times m-reduction [i] would yield digital (56, 161, 204)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3161, 204, F3, 105) (dual of [204, 43, 106]-code), but
- residual code [i] would yield OA(356, 98, S3, 35), but
- the linear programming bound shows that M ≥ 17594 267310 234640 548980 678885 211362 293573 050792 059460 366181 760458 388479 597863 317354 958813 552441 815811 709869 / 31 435807 153094 400646 640285 665269 261214 518362 272160 890223 276799 339913 769425 758125 > 356 [i]
- residual code [i] would yield OA(356, 98, S3, 35), but
- extracting embedded orthogonal array [i] would yield linear OA(3161, 204, F3, 105) (dual of [204, 43, 106]-code), but
(56, 56+106, 246)-Net in Base 3 — Upper bound on s
There is no (56, 162, 247)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 198903 213154 698696 837296 229875 852907 966373 922746 939006 333804 121474 819183 948871 > 3162 [i]