Best Known (32−16, 32, s)-Nets in Base 256
(32−16, 32, 8192)-Net over F256 — Constructive and digital
Digital (16, 32, 8192)-net over F256, using
- 1 times m-reduction [i] based on digital (16, 33, 8192)-net over F256, using
- net defined by OOA [i] based on linear OOA(25633, 8192, F256, 17, 17) (dual of [(8192, 17), 139231, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(25633, 65537, F256, 17) (dual of [65537, 65504, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- OOA 8-folding and stacking with additional row [i] based on linear OA(25633, 65537, F256, 17) (dual of [65537, 65504, 18]-code), using
- net defined by OOA [i] based on linear OOA(25633, 8192, F256, 17, 17) (dual of [(8192, 17), 139231, 18]-NRT-code), using
(32−16, 32, 16385)-Net over F256 — Digital
Digital (16, 32, 16385)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25632, 16385, F256, 4, 16) (dual of [(16385, 4), 65508, 17]-NRT-code), using
- OOA 4-folding [i] based on linear OA(25632, 65540, F256, 16) (dual of [65540, 65508, 17]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(25632, 65541, F256, 16) (dual of [65541, 65509, 17]-code), using
- OOA 4-folding [i] based on linear OA(25632, 65540, F256, 16) (dual of [65540, 65508, 17]-code), using
(32−16, 32, large)-Net in Base 256 — Upper bound on s
There is no (16, 32, large)-net in base 256, because
- 14 times m-reduction [i] would yield (16, 18, large)-net in base 256, but