Best Known (237−173, 237, s)-Nets in Base 3
(237−173, 237, 48)-Net over F3 — Constructive and digital
Digital (64, 237, 48)-net over F3, using
- t-expansion [i] based on digital (45, 237, 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
(237−173, 237, 64)-Net over F3 — Digital
Digital (64, 237, 64)-net over F3, using
- t-expansion [i] based on digital (49, 237, 64)-net over F3, using
- net from sequence [i] based on digital (49, 63)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 49 and N(F) ≥ 64, using
- net from sequence [i] based on digital (49, 63)-sequence over F3, using
(237−173, 237, 202)-Net over F3 — Upper bound on s (digital)
There is no digital (64, 237, 203)-net over F3, because
- 44 times m-reduction [i] would yield digital (64, 193, 203)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3193, 203, F3, 129) (dual of [203, 10, 130]-code), but
- construction Y1 [i] would yield
- linear OA(3192, 199, F3, 129) (dual of [199, 7, 130]-code), but
- residual code [i] would yield linear OA(363, 69, F3, 43) (dual of [69, 6, 44]-code), but
- OA(310, 203, S3, 4), but
- discarding factors would yield OA(310, 172, S3, 4), but
- the Rao or (dual) Hamming bound shows that M ≥ 59169 > 310 [i]
- discarding factors would yield OA(310, 172, S3, 4), but
- linear OA(3192, 199, F3, 129) (dual of [199, 7, 130]-code), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(3193, 203, F3, 129) (dual of [203, 10, 130]-code), but
(237−173, 237, 205)-Net in Base 3 — Upper bound on s
There is no (64, 237, 206)-net in base 3, because
- 36 times m-reduction [i] would yield (64, 201, 206)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3201, 206, S3, 137), but
- the (dual) Plotkin bound shows that M ≥ 21 514733 098945 856316 441287 084898 149773 167909 664924 954173 940571 900866 491763 005476 822545 160622 564081 / 23 > 3201 [i]
- extracting embedded orthogonal array [i] would yield OA(3201, 206, S3, 137), but