Best Known (48, 48+17, s)-Nets in Base 256
(48, 48+17, 1064960)-Net over F256 — Constructive and digital
Digital (48, 65, 1064960)-net over F256, using
- (u, u+v)-construction [i] based on
- digital (8, 16, 16385)-net over F256, using
- net defined by OOA [i] based on linear OOA(25616, 16385, F256, 8, 8) (dual of [(16385, 8), 131064, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(25616, 65540, F256, 8) (dual of [65540, 65524, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(25616, 65541, F256, 8) (dual of [65541, 65525, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(25615, 65536, F256, 8) (dual of [65536, 65521, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(25611, 65536, F256, 6) (dual of [65536, 65525, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(7) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(25616, 65541, F256, 8) (dual of [65541, 65525, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(25616, 65540, F256, 8) (dual of [65540, 65524, 9]-code), using
- net defined by OOA [i] based on linear OOA(25616, 16385, F256, 8, 8) (dual of [(16385, 8), 131064, 9]-NRT-code), using
- digital (32, 49, 1048575)-net over F256, using
- net defined by OOA [i] based on linear OOA(25649, 1048575, F256, 17, 17) (dual of [(1048575, 17), 17825726, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(25649, 8388601, F256, 17) (dual of [8388601, 8388552, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(25649, large, F256, 17) (dual of [large, large−49, 18]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(25649, large, F256, 17) (dual of [large, large−49, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(25649, 8388601, F256, 17) (dual of [8388601, 8388552, 18]-code), using
- net defined by OOA [i] based on linear OOA(25649, 1048575, F256, 17, 17) (dual of [(1048575, 17), 17825726, 18]-NRT-code), using
- digital (8, 16, 16385)-net over F256, using
(48, 48+17, large)-Net over F256 — Digital
Digital (48, 65, large)-net over F256, using
- t-expansion [i] based on digital (47, 65, large)-net over F256, using
- 3 times m-reduction [i] based on digital (47, 68, large)-net over F256, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(25668, large, F256, 21) (dual of [large, large−68, 22]-code), using
- 7 times code embedding in larger space [i] based on linear OA(25661, large, F256, 21) (dual of [large, large−61, 22]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [i]
- 7 times code embedding in larger space [i] based on linear OA(25661, large, F256, 21) (dual of [large, large−61, 22]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(25668, large, F256, 21) (dual of [large, large−68, 22]-code), using
- 3 times m-reduction [i] based on digital (47, 68, large)-net over F256, using
(48, 48+17, large)-Net in Base 256 — Upper bound on s
There is no (48, 65, large)-net in base 256, because
- 15 times m-reduction [i] would yield (48, 50, large)-net in base 256, but