Best Known (41, 41+121, s)-Nets in Base 4
(41, 41+121, 56)-Net over F4 — Constructive and digital
Digital (41, 162, 56)-net over F4, using
- t-expansion [i] based on digital (33, 162, 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
(41, 41+121, 75)-Net over F4 — Digital
Digital (41, 162, 75)-net over F4, using
- t-expansion [i] based on digital (40, 162, 75)-net over F4, using
- net from sequence [i] based on digital (40, 74)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 40 and N(F) ≥ 75, using
- net from sequence [i] based on digital (40, 74)-sequence over F4, using
(41, 41+121, 185)-Net over F4 — Upper bound on s (digital)
There is no digital (41, 162, 186)-net over F4, because
- 1 times m-reduction [i] would yield digital (41, 161, 186)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4161, 186, F4, 120) (dual of [186, 25, 121]-code), but
- construction Y1 [i] would yield
- linear OA(4160, 172, F4, 120) (dual of [172, 12, 121]-code), but
- construction Y1 [i] would yield
- linear OA(4159, 166, F4, 120) (dual of [166, 7, 121]-code), but
- residual code [i] would yield linear OA(439, 45, F4, 30) (dual of [45, 6, 31]-code), but
- OA(412, 172, S4, 6), but
- discarding factors would yield OA(412, 156, S4, 6), but
- the Rao or (dual) Hamming bound shows that M ≥ 16 866019 > 412 [i]
- discarding factors would yield OA(412, 156, S4, 6), but
- linear OA(4159, 166, F4, 120) (dual of [166, 7, 121]-code), but
- construction Y1 [i] would yield
- OA(425, 186, S4, 14), but
- discarding factors would yield OA(425, 162, S4, 14), but
- the Rao or (dual) Hamming bound shows that M ≥ 1131 065428 484512 > 425 [i]
- discarding factors would yield OA(425, 162, S4, 14), but
- linear OA(4160, 172, F4, 120) (dual of [172, 12, 121]-code), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(4161, 186, F4, 120) (dual of [186, 25, 121]-code), but
(41, 41+121, 272)-Net in Base 4 — Upper bound on s
There is no (41, 162, 273)-net in base 4, because
- 1 times m-reduction [i] would yield (41, 161, 273)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 9 972059 416038 090367 909591 332213 327328 350705 691596 195447 740617 211883 939692 608759 124084 549085 307456 > 4161 [i]