Best Known (53−14, 53, s)-Nets in Base 256
(53−14, 53, 1220216)-Net over F256 — Constructive and digital
Digital (39, 53, 1220216)-net over F256, using
- (u, u+v)-construction [i] based on
- digital (6, 13, 21845)-net over F256, using
- net defined by OOA [i] based on linear OOA(25613, 21845, F256, 7, 7) (dual of [(21845, 7), 152902, 8]-NRT-code), using
- appending kth column [i] based on linear OOA(25613, 21845, F256, 6, 7) (dual of [(21845, 6), 131057, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(25613, 65536, F256, 7) (dual of [65536, 65523, 8]-code), using
- an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- OOA 3-folding and stacking with additional row [i] based on linear OA(25613, 65536, F256, 7) (dual of [65536, 65523, 8]-code), using
- appending kth column [i] based on linear OOA(25613, 21845, F256, 6, 7) (dual of [(21845, 6), 131057, 8]-NRT-code), using
- net defined by OOA [i] based on linear OOA(25613, 21845, F256, 7, 7) (dual of [(21845, 7), 152902, 8]-NRT-code), using
- digital (26, 40, 1198371)-net over F256, using
- net defined by OOA [i] based on linear OOA(25640, 1198371, F256, 14, 14) (dual of [(1198371, 14), 16777154, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(25640, 8388597, F256, 14) (dual of [8388597, 8388557, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(25640, large, F256, 14) (dual of [large, large−40, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(25640, large, F256, 14) (dual of [large, large−40, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(25640, 8388597, F256, 14) (dual of [8388597, 8388557, 15]-code), using
- net defined by OOA [i] based on linear OOA(25640, 1198371, F256, 14, 14) (dual of [(1198371, 14), 16777154, 15]-NRT-code), using
- digital (6, 13, 21845)-net over F256, using
(53−14, 53, large)-Net over F256 — Digital
Digital (39, 53, large)-net over F256, using
- t-expansion [i] based on digital (38, 53, large)-net over F256, using
- 2 times m-reduction [i] based on digital (38, 55, large)-net over F256, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(25655, large, F256, 17) (dual of [large, large−55, 18]-code), using
- strength reduction [i] based on linear OA(25655, large, F256, 19) (dual of [large, large−55, 20]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- strength reduction [i] based on linear OA(25655, large, F256, 19) (dual of [large, large−55, 20]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(25655, large, F256, 17) (dual of [large, large−55, 18]-code), using
- 2 times m-reduction [i] based on digital (38, 55, large)-net over F256, using
(53−14, 53, large)-Net in Base 256 — Upper bound on s
There is no (39, 53, large)-net in base 256, because
- 12 times m-reduction [i] would yield (39, 41, large)-net in base 256, but