Best Known (87−24, 87, s)-Nets in Base 16
(87−24, 87, 1285)-Net over F16 — Constructive and digital
Digital (63, 87, 1285)-net over F16, using
- generalized (u, u+v)-construction [i] based on
- digital (7, 15, 257)-net over F16, using
- base reduction for projective spaces (embedding PG(7,256) in PG(14,16)) for nets [i] based on digital (0, 8, 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
- base reduction for projective spaces (embedding PG(7,256) in PG(14,16)) for nets [i] based on digital (0, 8, 257)-net over F256, using
- digital (12, 24, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 12, 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, 12, 257)-net over F256, using
- digital (24, 48, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 24, 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, 24, 257)-net over F256, using
- digital (7, 15, 257)-net over F16, using
(87−24, 87, 1366)-Net in Base 16 — Constructive
(63, 87, 1366)-net in base 16, using
- 161 times duplication [i] based on (62, 86, 1366)-net in base 16, using
- net defined by OOA [i] based on OOA(1686, 1366, S16, 24, 24), using
- OA 12-folding and stacking [i] based on OA(1686, 16392, S16, 24), using
- discarding parts of the base [i] based on linear OA(12849, 16392, F128, 24) (dual of [16392, 16343, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(20) [i] based on
- linear OA(12847, 16384, F128, 24) (dual of [16384, 16337, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(12841, 16384, F128, 21) (dual of [16384, 16343, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(1282, 8, F128, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,128)), using
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- Reed–Solomon code RS(126,128) [i]
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- construction X applied to Ce(23) ⊂ Ce(20) [i] based on
- discarding parts of the base [i] based on linear OA(12849, 16392, F128, 24) (dual of [16392, 16343, 25]-code), using
- OA 12-folding and stacking [i] based on OA(1686, 16392, S16, 24), using
- net defined by OOA [i] based on OOA(1686, 1366, S16, 24, 24), using
(87−24, 87, 22558)-Net over F16 — Digital
Digital (63, 87, 22558)-net over F16, using
(87−24, 87, large)-Net in Base 16 — Upper bound on s
There is no (63, 87, large)-net in base 16, because
- 22 times m-reduction [i] would yield (63, 65, large)-net in base 16, but