Best Known (54−15, 54, s)-Nets in Base 256
(54−15, 54, 1198886)-Net over F256 — Constructive and digital
Digital (39, 54, 1198886)-net over F256, using
- (u, u+v)-construction [i] based on
- digital (4, 11, 515)-net over F256, using
- net defined by OOA [i] based on linear OOA(25611, 515, F256, 7, 7) (dual of [(515, 7), 3594, 8]-NRT-code), using
- appending kth column [i] based on linear OOA(25611, 515, F256, 6, 7) (dual of [(515, 6), 3079, 8]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(2563, 257, F256, 6, 3) (dual of [(257, 6), 1539, 4]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(6;1539,256) [i]
- linear OOA(2568, 258, F256, 6, 7) (dual of [(258, 6), 1540, 8]-NRT-code), using
- extracting embedded OOA [i] based on digital (1, 8, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- extracting embedded OOA [i] based on digital (1, 8, 258)-net over F256, using
- linear OOA(2563, 257, F256, 6, 3) (dual of [(257, 6), 1539, 4]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(25611, 515, F256, 6, 7) (dual of [(515, 6), 3079, 8]-NRT-code), using
- net defined by OOA [i] based on linear OOA(25611, 515, F256, 7, 7) (dual of [(515, 7), 3594, 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 (4, 11, 515)-net over F256, using
(54−15, 54, large)-Net over F256 — Digital
Digital (39, 54, large)-net over F256, using
- t-expansion [i] based on digital (38, 54, large)-net over F256, using
- 1 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
- 1 times m-reduction [i] based on digital (38, 55, large)-net over F256, using
(54−15, 54, large)-Net in Base 256 — Upper bound on s
There is no (39, 54, large)-net in base 256, because
- 13 times m-reduction [i] would yield (39, 41, large)-net in base 256, but