Best Known (225−153, 225, s)-Nets in Base 3
(225−153, 225, 48)-Net over F3 — Constructive and digital
Digital (72, 225, 48)-net over F3, using
- t-expansion [i] based on digital (45, 225, 48)-net over F3, using
- net from sequence [i] based on digital (45, 47)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 45 and N(F) ≥ 48, using
- net from sequence [i] based on digital (45, 47)-sequence over F3, using
(225−153, 225, 84)-Net over F3 — Digital
Digital (72, 225, 84)-net over F3, using
- t-expansion [i] based on digital (71, 225, 84)-net over F3, using
- net from sequence [i] based on digital (71, 83)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 71 and N(F) ≥ 84, using
- net from sequence [i] based on digital (71, 83)-sequence over F3, using
(225−153, 225, 226)-Net over F3 — Upper bound on s (digital)
There is no digital (72, 225, 227)-net over F3, because
- 6 times m-reduction [i] would yield digital (72, 219, 227)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3219, 227, F3, 147) (dual of [227, 8, 148]-code), but
- residual code [i] would yield linear OA(372, 79, F3, 49) (dual of [79, 7, 50]-code), but
- 1 times truncation [i] would yield linear OA(371, 78, F3, 48) (dual of [78, 7, 49]-code), but
- residual code [i] would yield linear OA(323, 29, F3, 16) (dual of [29, 6, 17]-code), but
- 1 times truncation [i] would yield linear OA(322, 28, F3, 15) (dual of [28, 6, 16]-code), but
- “HHM†bound on codes from Brouwer’s database [i]
- 1 times truncation [i] would yield linear OA(322, 28, F3, 15) (dual of [28, 6, 16]-code), but
- residual code [i] would yield linear OA(323, 29, F3, 16) (dual of [29, 6, 17]-code), but
- 1 times truncation [i] would yield linear OA(371, 78, F3, 48) (dual of [78, 7, 49]-code), but
- residual code [i] would yield linear OA(372, 79, F3, 49) (dual of [79, 7, 50]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(3219, 227, F3, 147) (dual of [227, 8, 148]-code), but
(225−153, 225, 229)-Net in Base 3 — Upper bound on s
There is no (72, 225, 230)-net in base 3, because
- extracting embedded orthogonal array [i] would yield OA(3225, 230, S3, 153), but
- the (dual) Plotkin bound shows that M ≥ 18 229188 289943 120727 504416 312516 508406 590984 052945 304492 988605 913605 329052 670723 967367 455215 961400 359649 216883 / 77 > 3225 [i]