Best Known (41, 41+13, s)-Nets in Base 256
(41, 41+13, 2796200)-Net over F256 — Constructive and digital
Digital (41, 54, 2796200)-net over F256, using
- 2561 times duplication [i] based on digital (40, 53, 2796200)-net over F256, using
- net defined by OOA [i] based on linear OOA(25653, 2796200, F256, 14, 13) (dual of [(2796200, 14), 39146747, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(25653, 8388601, F256, 2, 13) (dual of [(8388601, 2), 16777149, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(25653, 8388602, F256, 2, 13) (dual of [(8388602, 2), 16777151, 14]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(25616, 4194301, F256, 2, 6) (dual of [(4194301, 2), 8388586, 7]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25616, 8388602, F256, 6) (dual of [8388602, 8388586, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(25616, large, F256, 6) (dual of [large, large−16, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(25616, large, F256, 6) (dual of [large, large−16, 7]-code), using
- OOA 2-folding [i] based on linear OA(25616, 8388602, F256, 6) (dual of [8388602, 8388586, 7]-code), using
- linear OOA(25637, 4194301, F256, 2, 13) (dual of [(4194301, 2), 8388565, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25637, 8388602, F256, 13) (dual of [8388602, 8388565, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(25637, large, F256, 13) (dual of [large, large−37, 14]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- discarding factors / shortening the dual code based on linear OA(25637, large, F256, 13) (dual of [large, large−37, 14]-code), using
- OOA 2-folding [i] based on linear OA(25637, 8388602, F256, 13) (dual of [8388602, 8388565, 14]-code), using
- linear OOA(25616, 4194301, F256, 2, 6) (dual of [(4194301, 2), 8388586, 7]-NRT-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OOA(25653, 8388602, F256, 2, 13) (dual of [(8388602, 2), 16777151, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(25653, 8388601, F256, 2, 13) (dual of [(8388601, 2), 16777149, 14]-NRT-code), using
- net defined by OOA [i] based on linear OOA(25653, 2796200, F256, 14, 13) (dual of [(2796200, 14), 39146747, 14]-NRT-code), using
(41, 41+13, large)-Net over F256 — Digital
Digital (41, 54, large)-net over F256, using
- t-expansion [i] based on digital (40, 54, large)-net over F256, using
- 4 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
- 4 times m-reduction [i] based on digital (40, 58, large)-net over F256, using
(41, 41+13, large)-Net in Base 256 — Upper bound on s
There is no (41, 54, large)-net in base 256, because
- 11 times m-reduction [i] would yield (41, 43, large)-net in base 256, but