Best Known (203−155, 203, s)-Nets in Base 4
(203−155, 203, 56)-Net over F4 — Constructive and digital
Digital (48, 203, 56)-net over F4, using
- t-expansion [i] based on digital (33, 203, 56)-net over F4, using
- net from sequence [i] based on digital (33, 55)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 33 and N(F) ≥ 56, using
- F5 from the tower of function fields by GarcÃa and Stichtenoth over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 33 and N(F) ≥ 56, using
- net from sequence [i] based on digital (33, 55)-sequence over F4, using
(203−155, 203, 81)-Net over F4 — Digital
Digital (48, 203, 81)-net over F4, using
- t-expansion [i] based on digital (46, 203, 81)-net over F4, using
- net from sequence [i] based on digital (46, 80)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 46 and N(F) ≥ 81, using
- net from sequence [i] based on digital (46, 80)-sequence over F4, using
(203−155, 203, 199)-Net over F4 — Upper bound on s (digital)
There is no digital (48, 203, 200)-net over F4, because
- 11 times m-reduction [i] would yield digital (48, 192, 200)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4192, 200, F4, 144) (dual of [200, 8, 145]-code), but
- construction Y1 [i] would yield
- linear OA(4191, 196, F4, 144) (dual of [196, 5, 145]-code), but
- residual code [i] would yield linear OA(447, 51, F4, 36) (dual of [51, 4, 37]-code), but
- OA(48, 200, S4, 4), but
- discarding factors would yield OA(48, 121, S4, 4), but
- the Rao or (dual) Hamming bound shows that M ≥ 65704 > 48 [i]
- discarding factors would yield OA(48, 121, S4, 4), but
- linear OA(4191, 196, F4, 144) (dual of [196, 5, 145]-code), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(4192, 200, F4, 144) (dual of [200, 8, 145]-code), but
(203−155, 203, 206)-Net in Base 4 — Upper bound on s
There is no (48, 203, 207)-net in base 4, because
- extracting embedded orthogonal array [i] would yield OA(4203, 207, S4, 155), but
- the (dual) Plotkin bound shows that M ≥ 2644 223875 160994 395807 661232 131084 159313 618731 857124 877138 595181 097623 164945 245383 300756 841758 861139 390364 848100 093433 217024 / 13 > 4203 [i]