Best Known (58, 58+23, s)-Nets in Base 16
(58, 58+23, 1285)-Net over F16 — Constructive and digital
Digital (58, 81, 1285)-net over F16, using
- generalized (u, u+v)-construction [i] based on
- digital (6, 13, 257)-net over F16, using
- base reduction for projective spaces (embedding PG(6,256) in PG(12,16)) 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, 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(6,256) in PG(12,16)) for nets [i] based on digital (0, 7, 257)-net over F256, using
- digital (11, 22, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 11, 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, 11, 257)-net over F256, using
- digital (23, 46, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 23, 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, 23, 257)-net over F256, using
- digital (6, 13, 257)-net over F16, using
(58, 58+23, 1489)-Net in Base 16 — Constructive
(58, 81, 1489)-net in base 16, using
- 161 times duplication [i] based on (57, 80, 1489)-net in base 16, using
- base change [i] based on (41, 64, 1489)-net in base 32, using
- 321 times duplication [i] based on (40, 63, 1489)-net in base 32, using
- base change [i] based on digital (22, 45, 1489)-net over F128, using
- net defined by OOA [i] based on linear OOA(12845, 1489, F128, 23, 23) (dual of [(1489, 23), 34202, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(12845, 16380, F128, 23) (dual of [16380, 16335, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(12845, 16384, F128, 23) (dual of [16384, 16339, 24]-code), using
- an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- discarding factors / shortening the dual code based on linear OA(12845, 16384, F128, 23) (dual of [16384, 16339, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(12845, 16380, F128, 23) (dual of [16380, 16335, 24]-code), using
- net defined by OOA [i] based on linear OOA(12845, 1489, F128, 23, 23) (dual of [(1489, 23), 34202, 24]-NRT-code), using
- base change [i] based on digital (22, 45, 1489)-net over F128, using
- 321 times duplication [i] based on (40, 63, 1489)-net in base 32, using
- base change [i] based on (41, 64, 1489)-net in base 32, using
(58, 58+23, 16383)-Net over F16 — Digital
Digital (58, 81, 16383)-net over F16, using
(58, 58+23, large)-Net in Base 16 — Upper bound on s
There is no (58, 81, large)-net in base 16, because
- 21 times m-reduction [i] would yield (58, 60, large)-net in base 16, but