Best Known (48, 48+16, s)-Nets in Base 16
(48, 48+16, 16385)-Net over F16 — Constructive and digital
Digital (48, 64, 16385)-net over F16, using
- net defined by OOA [i] based on linear OOA(1664, 16385, F16, 16, 16) (dual of [(16385, 16), 262096, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(1664, 131080, F16, 16) (dual of [131080, 131016, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(1664, 131082, F16, 16) (dual of [131082, 131018, 17]-code), using
- trace code [i] based on linear OA(25632, 65541, F256, 16) (dual of [65541, 65509, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- linear OA(25631, 65536, F256, 16) (dual of [65536, 65505, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(25627, 65536, F256, 14) (dual of [65536, 65509, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(2561, 5, F256, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(2561, s, F256, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- trace code [i] based on linear OA(25632, 65541, F256, 16) (dual of [65541, 65509, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(1664, 131082, F16, 16) (dual of [131082, 131018, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(1664, 131080, F16, 16) (dual of [131080, 131016, 17]-code), using
(48, 48+16, 105652)-Net over F16 — Digital
Digital (48, 64, 105652)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1664, 105652, F16, 16) (dual of [105652, 105588, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(1664, 131082, F16, 16) (dual of [131082, 131018, 17]-code), using
- trace code [i] based on linear OA(25632, 65541, F256, 16) (dual of [65541, 65509, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- linear OA(25631, 65536, F256, 16) (dual of [65536, 65505, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(25627, 65536, F256, 14) (dual of [65536, 65509, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(2561, 5, F256, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(2561, s, F256, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- trace code [i] based on linear OA(25632, 65541, F256, 16) (dual of [65541, 65509, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(1664, 131082, F16, 16) (dual of [131082, 131018, 17]-code), using
(48, 48+16, large)-Net in Base 16 — Upper bound on s
There is no (48, 64, large)-net in base 16, because
- 14 times m-reduction [i] would yield (48, 50, large)-net in base 16, but