Best Known (40−20, 40, s)-Nets in Base 256
(40−20, 40, 6554)-Net over F256 — Constructive and digital
Digital (20, 40, 6554)-net over F256, using
- net defined by OOA [i] based on linear OOA(25640, 6554, F256, 20, 20) (dual of [(6554, 20), 131040, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(25640, 65540, F256, 20) (dual of [65540, 65500, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(25640, 65541, F256, 20) (dual of [65541, 65501, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(17) [i] based on
- linear OA(25639, 65536, F256, 20) (dual of [65536, 65497, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(25635, 65536, F256, 18) (dual of [65536, 65501, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(19) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(25640, 65541, F256, 20) (dual of [65541, 65501, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(25640, 65540, F256, 20) (dual of [65540, 65500, 21]-code), using
(40−20, 40, 15164)-Net over F256 — Digital
Digital (20, 40, 15164)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25640, 15164, F256, 4, 20) (dual of [(15164, 4), 60616, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(25640, 16385, F256, 4, 20) (dual of [(16385, 4), 65500, 21]-NRT-code), using
- OOA 4-folding [i] based on linear OA(25640, 65540, F256, 20) (dual of [65540, 65500, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(25640, 65541, F256, 20) (dual of [65541, 65501, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(17) [i] based on
- linear OA(25639, 65536, F256, 20) (dual of [65536, 65497, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(25635, 65536, F256, 18) (dual of [65536, 65501, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(19) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(25640, 65541, F256, 20) (dual of [65541, 65501, 21]-code), using
- OOA 4-folding [i] based on linear OA(25640, 65540, F256, 20) (dual of [65540, 65500, 21]-code), using
- discarding factors / shortening the dual code based on linear OOA(25640, 16385, F256, 4, 20) (dual of [(16385, 4), 65500, 21]-NRT-code), using
(40−20, 40, large)-Net in Base 256 — Upper bound on s
There is no (20, 40, large)-net in base 256, because
- 18 times m-reduction [i] would yield (20, 22, large)-net in base 256, but