Best Known (139−110, 139, s)-Nets in Base 3
(139−110, 139, 37)-Net over F3 — Constructive and digital
Digital (29, 139, 37)-net over F3, using
- t-expansion [i] based on digital (27, 139, 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]
- net from sequence [i] based on digital (27, 36)-sequence over F3, using
(139−110, 139, 42)-Net over F3 — Digital
Digital (29, 139, 42)-net over F3, using
- net from sequence [i] based on digital (29, 41)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 29 and N(F) ≥ 42, using
(139−110, 139, 95)-Net over F3 — Upper bound on s (digital)
There is no digital (29, 139, 96)-net over F3, because
- 50 times m-reduction [i] would yield digital (29, 89, 96)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(389, 96, F3, 60) (dual of [96, 7, 61]-code), but
- residual code [i] would yield linear OA(329, 35, F3, 20) (dual of [35, 6, 21]-code), but
- “Bou†bound on codes from Brouwer’s database [i]
- residual code [i] would yield linear OA(329, 35, F3, 20) (dual of [35, 6, 21]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(389, 96, F3, 60) (dual of [96, 7, 61]-code), but
(139−110, 139, 96)-Net in Base 3 — Upper bound on s
There is no (29, 139, 97)-net in base 3, because
- 50 times m-reduction [i] would yield (29, 89, 97)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(389, 97, S3, 60), but
- the linear programming bound shows that M ≥ 532344 681908 373843 992394 006465 724030 651148 098857 / 151585 > 389 [i]
- extracting embedded orthogonal array [i] would yield OA(389, 97, S3, 60), but