Best Known (68−54, 68, s)-Nets in Base 3
(68−54, 68, 24)-Net over F3 — Constructive and digital
Digital (14, 68, 24)-net over F3, using
- t-expansion [i] based on digital (13, 68, 24)-net over F3, using
- net from sequence [i] based on digital (13, 23)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 13 and N(F) ≥ 24, using
- net from sequence [i] based on digital (13, 23)-sequence over F3, using
(68−54, 68, 49)-Net over F3 — Upper bound on s (digital)
There is no digital (14, 68, 50)-net over F3, because
- 24 times m-reduction [i] would yield digital (14, 44, 50)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(344, 50, F3, 30) (dual of [50, 6, 31]-code), but
- “HJL†bound on codes from Brouwer’s database [i]
- extracting embedded orthogonal array [i] would yield linear OA(344, 50, F3, 30) (dual of [50, 6, 31]-code), but
(68−54, 68, 51)-Net in Base 3 — Upper bound on s
There is no (14, 68, 52)-net in base 3, because
- 23 times m-reduction [i] would yield (14, 45, 52)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(345, 52, S3, 31), but
- the linear programming bound shows that M ≥ 239299 329230 617529 590083 / 76 > 345 [i]
- extracting embedded orthogonal array [i] would yield OA(345, 52, S3, 31), but