Best Known (54, 163, s)-Nets in Base 3
(54, 163, 48)-Net over F3 — Constructive and digital
Digital (54, 163, 48)-net over F3, using
- t-expansion [i] based on digital (45, 163, 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, 163, 64)-Net over F3 — Digital
Digital (54, 163, 64)-net over F3, using
- t-expansion [i] based on digital (49, 163, 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, 163, 170)-Net over F3 — Upper bound on s (digital)
There is no digital (54, 163, 171)-net over F3, because
- 1 times m-reduction [i] would yield digital (54, 162, 171)-net over F3, but
- extracting embedded orthogonal array [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
- extracting embedded orthogonal array [i] would yield linear OA(3162, 171, F3, 108) (dual of [171, 9, 109]-code), but
(54, 163, 233)-Net in Base 3 — Upper bound on s
There is no (54, 163, 234)-net in base 3, because
- 1 times m-reduction [i] would yield (54, 162, 234)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 221329 431746 114134 162782 399012 840356 673086 593912 339599 152867 464357 389327 386893 > 3162 [i]