Best Known (40, 160, s)-Nets in Base 4
(40, 160, 56)-Net over F4 — Constructive and digital
Digital (40, 160, 56)-net over F4, using
- t-expansion [i] based on digital (33, 160, 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
(40, 160, 75)-Net over F4 — Digital
Digital (40, 160, 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
(40, 160, 171)-Net over F4 — Upper bound on s (digital)
There is no digital (40, 160, 172)-net over F4, because
- extracting embedded orthogonal array [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
(40, 160, 264)-Net in Base 4 — Upper bound on s
There is no (40, 160, 265)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 2 153627 852648 971862 282865 424584 699451 257238 836756 826347 434727 870668 924638 433233 403750 780707 837888 > 4160 [i]