Best Known (38, 47, s)-Nets in Base 4
(38, 47, 4101)-Net over F4 — Constructive and digital
Digital (38, 47, 4101)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 5)-net over F4, using
- net from sequence [i] based on digital (0, 4)-sequence over F4, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 0 and N(F) ≥ 5, using
- the rational function field F4(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 4)-sequence over F4, using
- digital (34, 43, 4096)-net over F4, using
- net defined by OOA [i] based on linear OOA(443, 4096, F4, 9, 9) (dual of [(4096, 9), 36821, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(443, 16385, F4, 9) (dual of [16385, 16342, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- OOA 4-folding and stacking with additional row [i] based on linear OA(443, 16385, F4, 9) (dual of [16385, 16342, 10]-code), using
- net defined by OOA [i] based on linear OOA(443, 4096, F4, 9, 9) (dual of [(4096, 9), 36821, 10]-NRT-code), using
- digital (0, 4, 5)-net over F4, using
(38, 47, 10186)-Net over F4 — Digital
Digital (38, 47, 10186)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(447, 10186, F4, 9) (dual of [10186, 10139, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(447, 16390, F4, 9) (dual of [16390, 16343, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([1,4]) [i] based on
- linear OA(443, 16385, F4, 9) (dual of [16385, 16342, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(442, 16385, F4, 4) (dual of [16385, 16343, 5]-code), using the narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(44, 5, F4, 4) (dual of [5, 1, 5]-code or 5-arc in PG(3,4)), using
- dual of repetition code with length 5 [i]
- construction X applied to C([0,4]) ⊂ C([1,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(447, 16390, F4, 9) (dual of [16390, 16343, 10]-code), using
(38, 47, 6189011)-Net in Base 4 — Upper bound on s
There is no (38, 47, 6189012)-net in base 4, because
- 1 times m-reduction [i] would yield (38, 46, 6189012)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 4951 763240 034962 258325 624484 > 446 [i]