Best Known (56, 56+112, s)-Nets in Base 3
(56, 56+112, 48)-Net over F3 — Constructive and digital
Digital (56, 168, 48)-net over F3, using
- t-expansion [i] based on digital (45, 168, 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+112, 64)-Net over F3 — Digital
Digital (56, 168, 64)-net over F3, using
- t-expansion [i] based on digital (49, 168, 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+112, 180)-Net over F3 — Upper bound on s (digital)
There is no digital (56, 168, 181)-net over F3, because
- 1 times m-reduction [i] would yield digital (56, 167, 181)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3167, 181, F3, 111) (dual of [181, 14, 112]-code), but
- construction Y1 [i] would yield
- linear OA(3166, 175, F3, 111) (dual of [175, 9, 112]-code), but
- construction Y1 [i] would yield
- linear OA(3165, 171, F3, 111) (dual of [171, 6, 112]-code), but
- residual code [i] would yield linear OA(354, 59, F3, 37) (dual of [59, 5, 38]-code), but
- 1 times truncation [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
- 1 times truncation [i] would yield linear OA(353, 58, F3, 36) (dual of [58, 5, 37]-code), but
- residual code [i] would yield linear OA(354, 59, F3, 37) (dual of [59, 5, 38]-code), but
- OA(39, 175, 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(3165, 171, F3, 111) (dual of [171, 6, 112]-code), but
- construction Y1 [i] would yield
- OA(314, 181, 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(3166, 175, F3, 111) (dual of [175, 9, 112]-code), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(3167, 181, F3, 111) (dual of [181, 14, 112]-code), but
(56, 56+112, 241)-Net in Base 3 — Upper bound on s
There is no (56, 168, 242)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 156 391109 237959 143391 476341 767692 331117 790750 885142 929376 561213 177730 234343 042865 > 3168 [i]