Best Known (48−13, 48, s)-Nets in Base 16
(48−13, 48, 1544)-Net over F16 — Constructive and digital
Digital (35, 48, 1544)-net over F16, using
- generalized (u, u+v)-construction [i] based on
- digital (4, 8, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 4, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 4, 257)-net over F256, using
- digital (6, 12, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 6, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256 (see above)
- trace code for nets [i] based on digital (0, 6, 257)-net over F256, using
- digital (15, 28, 516)-net over F16, using
- trace code for nets [i] based on digital (1, 14, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- trace code for nets [i] based on digital (1, 14, 258)-net over F256, using
- digital (4, 8, 514)-net over F16, using
(48−13, 48, 5462)-Net in Base 16 — Constructive
(35, 48, 5462)-net in base 16, using
- net defined by OOA [i] based on OOA(1648, 5462, S16, 13, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(1648, 32773, S16, 13), using
- discarding factors based on OA(1648, 32776, S16, 13), using
- discarding parts of the base [i] based on linear OA(3238, 32776, F32, 13) (dual of [32776, 32738, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(3237, 32769, F32, 13) (dual of [32769, 32732, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(3231, 32769, F32, 11) (dual of [32769, 32738, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(321, 7, F32, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(321, s, F32, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- discarding parts of the base [i] based on linear OA(3238, 32776, F32, 13) (dual of [32776, 32738, 14]-code), using
- discarding factors based on OA(1648, 32776, S16, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(1648, 32773, S16, 13), using
(48−13, 48, 23113)-Net over F16 — Digital
Digital (35, 48, 23113)-net over F16, using
(48−13, 48, large)-Net in Base 16 — Upper bound on s
There is no (35, 48, large)-net in base 16, because
- 11 times m-reduction [i] would yield (35, 37, large)-net in base 16, but