Best Known (55−15, 55, s)-Nets in Base 16
(55−15, 55, 1542)-Net over F16 — Constructive and digital
Digital (40, 55, 1542)-net over F16, using
- 161 times duplication [i] based on digital (39, 54, 1542)-net over F16, using
- generalized (u, u+v)-construction [i] based on
- digital (5, 10, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 5, 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, 5, 257)-net over F256, using
- digital (7, 14, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 7, 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, 7, 257)-net over F256, using
- digital (15, 30, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 15, 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, 15, 257)-net over F256, using
- digital (5, 10, 514)-net over F16, using
- generalized (u, u+v)-construction [i] based on
(55−15, 55, 4682)-Net in Base 16 — Constructive
(40, 55, 4682)-net in base 16, using
- base change [i] based on digital (29, 44, 4682)-net over F32, using
- net defined by OOA [i] based on linear OOA(3244, 4682, F32, 15, 15) (dual of [(4682, 15), 70186, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(3244, 32775, F32, 15) (dual of [32775, 32731, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(3244, 32776, F32, 15) (dual of [32776, 32732, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(3243, 32769, F32, 15) (dual of [32769, 32726, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- 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(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([0,7]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3244, 32776, F32, 15) (dual of [32776, 32732, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(3244, 32775, F32, 15) (dual of [32775, 32731, 16]-code), using
- net defined by OOA [i] based on linear OOA(3244, 4682, F32, 15, 15) (dual of [(4682, 15), 70186, 16]-NRT-code), using
(55−15, 55, 21676)-Net over F16 — Digital
Digital (40, 55, 21676)-net over F16, using
(55−15, 55, large)-Net in Base 16 — Upper bound on s
There is no (40, 55, large)-net in base 16, because
- 13 times m-reduction [i] would yield (40, 42, large)-net in base 16, but