Best Known (53−17, 53, s)-Nets in Base 256
(53−17, 53, 1048575)-Net over F256 — Constructive and digital
Digital (36, 53, 1048575)-net over F256, using
- 2564 times duplication [i] based on 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
(53−17, 53, 5620710)-Net over F256 — Digital
Digital (36, 53, 5620710)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(25653, 5620710, F256, 17) (dual of [5620710, 5620657, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(25653, large, F256, 17) (dual of [large, large−53, 18]-code), using
- 4 times code embedding in larger space [i] 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]
- 4 times code embedding in larger space [i] based on linear OA(25649, large, F256, 17) (dual of [large, large−49, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(25653, large, F256, 17) (dual of [large, large−53, 18]-code), using
(53−17, 53, large)-Net in Base 256 — Upper bound on s
There is no (36, 53, large)-net in base 256, because
- 15 times m-reduction [i] would yield (36, 38, large)-net in base 256, but