Best Known (31−21, 31, s)-Nets in Base 3
(31−21, 31, 19)-Net over F3 — Constructive and digital
Digital (10, 31, 19)-net over F3, using
- t-expansion [i] based on digital (9, 31, 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
- net from sequence [i] based on digital (9, 18)-sequence over F3, using
(31−21, 31, 20)-Net over F3 — Digital
Digital (10, 31, 20)-net over F3, using
- net from sequence [i] based on digital (10, 19)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 10 and N(F) ≥ 20, using
(31−21, 31, 36)-Net over F3 — Upper bound on s (digital)
There is no digital (10, 31, 37)-net over F3, because
- extracting embedded orthogonal array [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]
(31−21, 31, 45)-Net in Base 3 — Upper bound on s
There is no (10, 31, 46)-net in base 3, because
- extracting embedded orthogonal array [i] would yield OA(331, 46, S3, 21), but
- the linear programming bound shows that M ≥ 65199 458532 028151 172069 / 101 781680 > 331 [i]