Best Known (54, 54+12, s)-Nets in Base 256
(54, 54+12, 5592401)-Net over F256 — Constructive and digital
Digital (54, 66, 5592401)-net over F256, using
- net defined by OOA [i] based on linear OOA(25666, 5592401, F256, 18, 12) (dual of [(5592401, 18), 100663152, 13]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(25666, 5592402, F256, 6, 12) (dual of [(5592402, 6), 33554346, 13]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(25616, 2796201, F256, 6, 6) (dual of [(2796201, 6), 16777190, 7]-NRT-code), using
- OA 3-folding and stacking [i] 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]
- OA 3-folding and stacking [i] based on linear OA(25616, large, F256, 6) (dual of [large, large−16, 7]-code), using
- linear OOA(25650, 2796201, F256, 6, 12) (dual of [(2796201, 6), 16777156, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OOA(25650, 5592402, F256, 3, 12) (dual of [(5592402, 3), 16777156, 13]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(25616, 2796201, F256, 3, 6) (dual of [(2796201, 3), 8388587, 7]-NRT-code), using
- OOA 3-folding [i] based on linear OA(25616, large, F256, 6) (dual of [large, large−16, 7]-code) (see above)
- linear OOA(25634, 2796201, F256, 3, 12) (dual of [(2796201, 3), 8388569, 13]-NRT-code), using
- OOA 3-folding [i] based on linear OA(25634, large, F256, 12) (dual of [large, large−34, 13]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- OOA 3-folding [i] based on linear OA(25634, large, F256, 12) (dual of [large, large−34, 13]-code), using
- linear OOA(25616, 2796201, F256, 3, 6) (dual of [(2796201, 3), 8388587, 7]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA 2-folding [i] based on linear OOA(25650, 5592402, F256, 3, 12) (dual of [(5592402, 3), 16777156, 13]-NRT-code), using
- linear OOA(25616, 2796201, F256, 6, 6) (dual of [(2796201, 6), 16777190, 7]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(25666, 5592402, F256, 6, 12) (dual of [(5592402, 6), 33554346, 13]-NRT-code), using
(54, 54+12, large)-Net over F256 — Digital
Digital (54, 66, large)-net over F256, using
- t-expansion [i] based on digital (47, 66, large)-net over F256, using
- 2 times m-reduction [i] based on digital (47, 68, large)-net over F256, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(25668, large, F256, 21) (dual of [large, large−68, 22]-code), using
- 7 times code embedding in larger space [i] based on linear OA(25661, large, F256, 21) (dual of [large, large−61, 22]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [i]
- 7 times code embedding in larger space [i] based on linear OA(25661, large, F256, 21) (dual of [large, large−61, 22]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(25668, large, F256, 21) (dual of [large, large−68, 22]-code), using
- 2 times m-reduction [i] based on digital (47, 68, large)-net over F256, using
(54, 54+12, large)-Net in Base 256 — Upper bound on s
There is no (54, 66, large)-net in base 256, because
- 10 times m-reduction [i] would yield (54, 56, large)-net in base 256, but