Best Known (12, 12+21, s)-Nets in Base 3
(12, 12+21, 20)-Net over F3 — Constructive and digital
Digital (12, 33, 20)-net over F3, using
- t-expansion [i] based on digital (11, 33, 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]
- net from sequence [i] based on digital (11, 19)-sequence over F3, using
(12, 12+21, 22)-Net over F3 — Digital
Digital (12, 33, 22)-net over F3, using
- net from sequence [i] based on digital (12, 21)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 12 and N(F) ≥ 22, using
(12, 12+21, 66)-Net over F3 — Upper bound on s (digital)
There is no digital (12, 33, 67)-net over F3, because
- extracting embedded orthogonal array [i] would yield linear OA(333, 67, F3, 21) (dual of [67, 34, 22]-code), but
- construction Y1 [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
- OA(334, 67, S3, 22), but
- discarding factors would yield OA(334, 64, S3, 22), but
- the linear programming bound shows that M ≥ 14 111105 234476 582146 780186 612485 543683 142605 088372 524627 579947 507340 583313 / 837 608500 983525 185318 819414 283441 298949 343873 086193 807233 > 334 [i]
- discarding factors would yield OA(334, 64, S3, 22), but
- linear OA(332, 45, F3, 21) (dual of [45, 13, 22]-code), but
- construction Y1 [i] would yield
(12, 12+21, 67)-Net in Base 3 — Upper bound on s
There is no (12, 33, 68)-net in base 3, because
- 1 times m-reduction [i] would yield (12, 32, 68)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 2077 150742 832697 > 332 [i]