Best Known (9, 9+18, s)-Nets in Base 3
(9, 9+18, 19)-Net over F3 — Constructive and digital
Digital (9, 27, 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
(9, 9+18, 34)-Net over F3 — Upper bound on s (digital)
There is no digital (9, 27, 35)-net over F3, because
- 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
(9, 9+18, 47)-Net in Base 3 — Upper bound on s
There is no (9, 27, 48)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 7 634663 630689 > 327 [i]
- extracting embedded orthogonal array [i] would yield OA(327, 48, S3, 18), but
- the linear programming bound shows that M ≥ 2857 592773 200213 559789 477301 061331 833249 / 360 761999 058230 313984 798589 > 327 [i]