Best Known (51−40, 51, s)-Nets in Base 3
(51−40, 51, 20)-Net over F3 — Constructive and digital
Digital (11, 51, 20)-net over F3, using
- net from sequence [i] based on digital (11, 19)-sequence over F3, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F3 with g(F) = 9, N(F) = 19, and 1 place with degree 3 [i] based on function field F/F3 with g(F) = 9 and N(F) ≥ 19, using an explicitly constructive algebraic function field [i]
(51−40, 51, 39)-Net over F3 — Upper bound on s (digital)
There is no digital (11, 51, 40)-net over F3, because
- 16 times m-reduction [i] would yield digital (11, 35, 40)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(335, 40, F3, 24) (dual of [40, 5, 25]-code), but
- “vE1†bound on codes from Brouwer’s database [i]
- extracting embedded orthogonal array [i] would yield linear OA(335, 40, F3, 24) (dual of [40, 5, 25]-code), but
(51−40, 51, 42)-Net in Base 3 — Upper bound on s
There is no (11, 51, 43)-net in base 3, because
- 11 times m-reduction [i] would yield (11, 40, 43)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(340, 43, S3, 29), but
- the (dual) Plotkin bound shows that M ≥ 72 945992 754341 572806 / 5 > 340 [i]
- extracting embedded orthogonal array [i] would yield OA(340, 43, S3, 29), but