Best Known (27, 27+56, s)-Nets in Base 3
(27, 27+56, 37)-Net over F3 — Constructive and digital
Digital (27, 83, 37)-net over F3, using
- net from sequence [i] based on digital (27, 36)-sequence over F3, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F3 with g(F) = 26, N(F) = 36, and 1 place with degree 2 [i] based on function field F/F3 with g(F) = 26 and N(F) ≥ 36, using an explicitly constructive algebraic function field [i]
(27, 27+56, 39)-Net over F3 — Digital
Digital (27, 83, 39)-net over F3, using
- net from sequence [i] based on digital (27, 38)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 27 and N(F) ≥ 39, using
(27, 27+56, 89)-Net over F3 — Upper bound on s (digital)
There is no digital (27, 83, 90)-net over F3, because
- 2 times m-reduction [i] would yield digital (27, 81, 90)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(381, 90, F3, 54) (dual of [90, 9, 55]-code), but
- residual code [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
- residual code [i] would yield linear OA(327, 35, F3, 18) (dual of [35, 8, 19]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(381, 90, F3, 54) (dual of [90, 9, 55]-code), but
(27, 27+56, 90)-Net in Base 3 — Upper bound on s
There is no (27, 83, 91)-net in base 3, because
- extracting embedded orthogonal array [i] would yield OA(383, 91, S3, 56), but
- the linear programming bound shows that M ≥ 969773 729787 523602 876821 942164 080815 560161 / 209 > 383 [i]