Best Known (165−109, 165, s)-Nets in Base 3
(165−109, 165, 48)-Net over F3 — Constructive and digital
Digital (56, 165, 48)-net over F3, using
- t-expansion [i] based on digital (45, 165, 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
(165−109, 165, 64)-Net over F3 — Digital
Digital (56, 165, 64)-net over F3, using
- t-expansion [i] based on digital (49, 165, 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
(165−109, 165, 188)-Net over F3 — Upper bound on s (digital)
There is no digital (56, 165, 189)-net over F3, because
- 1 times m-reduction [i] would yield digital (56, 164, 189)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3164, 189, F3, 108) (dual of [189, 25, 109]-code), but
- construction Y1 [i] would yield
- linear OA(3163, 177, F3, 108) (dual of [177, 14, 109]-code), but
- construction Y1 [i] would yield
- linear OA(3162, 171, F3, 108) (dual of [171, 9, 109]-code), but
- construction Y1 [i] would yield
- linear OA(3161, 167, F3, 108) (dual of [167, 6, 109]-code), but
- residual code [i] would yield linear OA(353, 58, F3, 36) (dual of [58, 5, 37]-code), but
- residual code [i] would yield linear OA(317, 21, F3, 12) (dual of [21, 4, 13]-code), but
- residual code [i] would yield linear OA(353, 58, F3, 36) (dual of [58, 5, 37]-code), but
- OA(39, 171, S3, 4), but
- discarding factors would yield OA(39, 100, S3, 4), but
- the Rao or (dual) Hamming bound shows that M ≥ 20001 > 39 [i]
- discarding factors would yield OA(39, 100, S3, 4), but
- linear OA(3161, 167, F3, 108) (dual of [167, 6, 109]-code), but
- construction Y1 [i] would yield
- OA(314, 177, S3, 6), but
- discarding factors would yield OA(314, 154, S3, 6), but
- the Rao or (dual) Hamming bound shows that M ≥ 4 822665 > 314 [i]
- discarding factors would yield OA(314, 154, S3, 6), but
- linear OA(3162, 171, F3, 108) (dual of [171, 9, 109]-code), but
- construction Y1 [i] would yield
- OA(325, 189, S3, 12), but
- discarding factors would yield OA(325, 148, S3, 12), but
- the Rao or (dual) Hamming bound shows that M ≥ 861032 991633 > 325 [i]
- discarding factors would yield OA(325, 148, S3, 12), but
- linear OA(3163, 177, F3, 108) (dual of [177, 14, 109]-code), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(3164, 189, F3, 108) (dual of [189, 25, 109]-code), but
(165−109, 165, 245)-Net in Base 3 — Upper bound on s
There is no (56, 165, 246)-net in base 3, because
- 1 times m-reduction [i] would yield (56, 164, 246)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 2 128449 665289 168784 465166 823345 005173 648527 963391 899232 492701 570260 580713 384661 > 3164 [i]