Best Known (65−33, 65, s)-Nets in Base 256
(65−33, 65, 4096)-Net over F256 — Constructive and digital
Digital (32, 65, 4096)-net over F256, using
- net defined by OOA [i] based on linear OOA(25665, 4096, F256, 33, 33) (dual of [(4096, 33), 135103, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(25665, 65537, F256, 33) (dual of [65537, 65472, 34]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- OOA 16-folding and stacking with additional row [i] based on linear OA(25665, 65537, F256, 33) (dual of [65537, 65472, 34]-code), using
(65−33, 65, 10923)-Net over F256 — Digital
Digital (32, 65, 10923)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25665, 10923, F256, 6, 33) (dual of [(10923, 6), 65473, 34]-NRT-code), using
- OOA 6-folding [i] based on linear OA(25665, 65538, F256, 33) (dual of [65538, 65473, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(31) [i] based on
- linear OA(25665, 65536, F256, 33) (dual of [65536, 65471, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(25663, 65536, F256, 32) (dual of [65536, 65473, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(32) ⊂ Ce(31) [i] based on
- OOA 6-folding [i] based on linear OA(25665, 65538, F256, 33) (dual of [65538, 65473, 34]-code), using
(65−33, 65, large)-Net in Base 256 — Upper bound on s
There is no (32, 65, large)-net in base 256, because
- 31 times m-reduction [i] would yield (32, 34, large)-net in base 256, but