Best Known (54, 54+103, s)-Nets in Base 3
(54, 54+103, 48)-Net over F3 — Constructive and digital
Digital (54, 157, 48)-net over F3, using
- t-expansion [i] based on digital (45, 157, 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+103, 64)-Net over F3 — Digital
Digital (54, 157, 64)-net over F3, using
- t-expansion [i] based on digital (49, 157, 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+103, 193)-Net over F3 — Upper bound on s (digital)
There is no digital (54, 157, 194)-net over F3, because
- 1 times m-reduction [i] would yield digital (54, 156, 194)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3156, 194, F3, 102) (dual of [194, 38, 103]-code), but
- residual code [i] would yield OA(354, 91, S3, 34), but
- the linear programming bound shows that M ≥ 474 977782 539634 784051 796341 060967 928894 025210 202817 107183 513784 170892 978866 150753 / 7 995136 378245 997436 849186 466159 148790 740510 898632 999345 > 354 [i]
- residual code [i] would yield OA(354, 91, S3, 34), but
- extracting embedded orthogonal array [i] would yield linear OA(3156, 194, F3, 102) (dual of [194, 38, 103]-code), but
(54, 54+103, 238)-Net in Base 3 — Upper bound on s
There is no (54, 157, 239)-net in base 3, because
- 1 times m-reduction [i] would yield (54, 156, 239)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 278 420477 880798 026422 004823 387841 349895 182663 771015 021451 685601 885338 777179 > 3156 [i]