Best Known (49, 49+92, s)-Nets in Base 3
(49, 49+92, 48)-Net over F3 — Constructive and digital
Digital (49, 141, 48)-net over F3, using
- t-expansion [i] based on digital (45, 141, 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
(49, 49+92, 64)-Net over F3 — Digital
Digital (49, 141, 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
(49, 49+92, 181)-Net over F3 — Upper bound on s (digital)
There is no digital (49, 141, 182)-net over F3, because
- 1 times m-reduction [i] would yield digital (49, 140, 182)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3140, 182, F3, 91) (dual of [182, 42, 92]-code), but
- construction Y1 [i] would yield
- linear OA(3139, 160, F3, 91) (dual of [160, 21, 92]-code), but
- construction Y1 [i] would yield
- OA(3138, 150, S3, 91), but
- the linear programming bound shows that M ≥ 13 106423 371120 876472 986931 421169 566720 811359 868949 528224 695915 065174 686944 / 17 599301 > 3138 [i]
- OA(321, 160, S3, 10), but
- discarding factors would yield OA(321, 133, S3, 10), but
- the Rao or (dual) Hamming bound shows that M ≥ 10487 287539 > 321 [i]
- discarding factors would yield OA(321, 133, S3, 10), but
- OA(3138, 150, S3, 91), but
- construction Y1 [i] would yield
- OA(342, 182, S3, 22), but
- discarding factors would yield OA(342, 168, S3, 22), but
- the Rao or (dual) Hamming bound shows that M ≥ 114 464714 711551 910433 > 342 [i]
- discarding factors would yield OA(342, 168, S3, 22), but
- linear OA(3139, 160, F3, 91) (dual of [160, 21, 92]-code), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(3140, 182, F3, 91) (dual of [182, 42, 92]-code), but
(49, 49+92, 218)-Net in Base 3 — Upper bound on s
There is no (49, 141, 219)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 20 368334 404390 959451 562640 977170 379786 696898 223629 539621 222167 819773 > 3141 [i]