Best Known (51−42, 51, s)-Nets in Base 3
(51−42, 51, 19)-Net over F3 — Constructive and digital
Digital (9, 51, 19)-net over F3, using
- net from sequence [i] based on digital (9, 18)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 9 and N(F) ≥ 19, using
(51−42, 51, 34)-Net over F3 — Upper bound on s (digital)
There is no digital (9, 51, 35)-net over F3, because
- 24 times m-reduction [i] would yield digital (9, 27, 35)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(327, 35, F3, 18) (dual of [35, 8, 19]-code), but
- residual code [i] would yield linear OA(39, 16, F3, 6) (dual of [16, 7, 7]-code), but
- “vE2†bound on codes from Brouwer’s database [i]
- residual code [i] would yield linear OA(39, 16, F3, 6) (dual of [16, 7, 7]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(327, 35, F3, 18) (dual of [35, 8, 19]-code), but
(51−42, 51, 35)-Net in Base 3 — Upper bound on s
There is no (9, 51, 36)-net in base 3, because
- 18 times m-reduction [i] would yield (9, 33, 36)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(333, 36, S3, 24), but
- the (dual) Plotkin bound shows that M ≥ 150094 635296 999121 / 25 > 333 [i]
- extracting embedded orthogonal array [i] would yield OA(333, 36, S3, 24), but