Best Known (41, 41+15, s)-Nets in Base 256
(41, 41+15, 1220216)-Net over F256 — Constructive and digital
Digital (41, 56, 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 (28, 43, 1198371)-net over F256, using
- net defined by OOA [i] based on linear OOA(25643, 1198371, F256, 15, 15) (dual of [(1198371, 15), 17975522, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(25643, 8388598, F256, 15) (dual of [8388598, 8388555, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(25643, large, F256, 15) (dual of [large, large−43, 16]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(25643, large, F256, 15) (dual of [large, large−43, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(25643, 8388598, F256, 15) (dual of [8388598, 8388555, 16]-code), using
- net defined by OOA [i] based on linear OOA(25643, 1198371, F256, 15, 15) (dual of [(1198371, 15), 17975522, 16]-NRT-code), using
- digital (6, 13, 21845)-net over F256, using
(41, 41+15, large)-Net over F256 — Digital
Digital (41, 56, large)-net over F256, using
- t-expansion [i] based on digital (40, 56, large)-net over F256, using
- 2 times m-reduction [i] based on digital (40, 58, large)-net over F256, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(25658, large, F256, 18) (dual of [large, large−58, 19]-code), using
- strength reduction [i] based on linear OA(25658, large, F256, 20) (dual of [large, large−58, 21]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [i]
- strength reduction [i] based on linear OA(25658, large, F256, 20) (dual of [large, large−58, 21]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(25658, large, F256, 18) (dual of [large, large−58, 19]-code), using
- 2 times m-reduction [i] based on digital (40, 58, large)-net over F256, using
(41, 41+15, large)-Net in Base 256 — Upper bound on s
There is no (41, 56, large)-net in base 256, because
- 13 times m-reduction [i] would yield (41, 43, large)-net in base 256, but