Best Known (11, 11+23, s)-Nets in Base 3
(11, 11+23, 20)-Net over F3 — Constructive and digital
Digital (11, 34, 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]
(11, 11+23, 44)-Net over F3 — Upper bound on s (digital)
There is no digital (11, 34, 45)-net over F3, because
- 2 times m-reduction [i] would yield digital (11, 32, 45)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(332, 45, F3, 21) (dual of [45, 13, 22]-code), but
- construction Y1 [i] would yield
- linear OA(331, 37, F3, 21) (dual of [37, 6, 22]-code), but
- “Bou†bound on codes from Brouwer’s database [i]
- OA(313, 45, S3, 8), but
- discarding factors would yield OA(313, 41, S3, 8), but
- the Rao or (dual) Hamming bound shows that M ≥ 1 708963 > 313 [i]
- discarding factors would yield OA(313, 41, S3, 8), but
- linear OA(331, 37, F3, 21) (dual of [37, 6, 22]-code), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(332, 45, F3, 21) (dual of [45, 13, 22]-code), but
(11, 11+23, 46)-Net in Base 3 — Upper bound on s
There is no (11, 34, 47)-net in base 3, because
- extracting embedded orthogonal array [i] would yield OA(334, 47, S3, 23), but
- the linear programming bound shows that M ≥ 766820 433499 097670 265473 / 42 619850 > 334 [i]