1996638
1558623
3052177
1134053
2925715
1027551
783007
520206
2662508
827121
1769631
3079690
461260
1082332
2542376
933073
1164221
3135973
1197852
25398
alpha = 999/1000
The unimodular matrix found is

 0 -1  0 -2  0  0  0  0  1  0  0  0  0  0  0  0  1  0  0  0 
 0  0  0 -1  0 -1 -1  0  1  1  0  1  0 -1 -1  0  0  0  0  0 
 0  0  1  0  0  0  0  0  0 -1 -1 -1  0 -1  1  0  1  0  0  0 
 0  0  0  0  0 -1  1  0  0  0 -1  1 -1  0 -1  0  0  1 -1  0 
 0  0 -1 -1  1 -1  0  0  1 -1  0  0  0  1 -1  0  0  1 -1 -1 
 0  1  0  1  0  0 -1 -1  0 -2  0  0  0  0  0 -1  0  0  1  0 
-1  1 -1  1  0  0 -1  0  0 -1  1  0 -1 -1  1  0  0  0  1  0 
 2 -1  0  0 -1  1  1  0  0  1  0 -1  0  0  0  1  0  0  0  0 
 0  0  1  1 -1 -1  0 -1  0  0  0  0 -1  0  0  0 -1  1 -1 -1 
 1  1  0 -1 -1  1  1  0  0  0 -1  0 -1  0  0  1  1  0 -1  1 
 1  0 -1 -1  1  0  1 -1  0 -1  0  1  0  1  0  0  0 -1 -1  0 
 0 -1  1  0  0 -1  0  0 -1 -2  1  0 -1  0  1  0  0  0  0  0 
 0 -1  0  1  0 -1  0  0  0  2  1  0  0  0  0  0  1 -1  0  0 
 0 -1  0  1  0  0 -1 -1  2  0  0  0 -1  0  0  0  0 -1  0  0 
 0 -1  1 -1  0  1  0 -1  0  0  1  0  1 -2  0 -1  0  0  0  0 
 0  0  0  0 -1 -2  1 -1  1  0  0  1  1  1 -1  0  0  0  0 -1 
 0  0  0  1  0  0 -1  0  0 -1 -1 -1  2  1  0  1  1  0  1  1 
 0  0  0  1  1 -1  1  0  0  0  2 -1  0  0  0  0 -1 -1  0  1 
-1  1 -1 -1 -1  0 -1  1  1  0  0  1  1  0  1 -1  0  0  0  0 
 0  0  1  0  0 -1  0  0  0  0  0  0  1 -1 -1  0  1  0  0 -1
Multiplier vector found by LLL is
b[20]=0 0 1 0 0 -1 0 0 0 0 0 0 1 -1 -1 0 1 0 0 -1 
Euclidean norm squared = 7. This is the shortest multiplier.

----------------------------------------------------------
alpha = 1
Unimodular matrix

 0 -1  0 -2  0  0  0  0  1  0  0  0  0  0  0  0  1  0  0  0 
 0  0  0 -1  0 -1 -1  0  1  1  0  1  0 -1 -1  0  0  0  0  0 
 0  0  1  0  0  0  0  0  0 -1 -1 -1  0 -1  1  0  1  0  0  0 
 0  0  0  0  0 -1  1  0  0  0 -1  1 -1  0 -1  0  0  1 -1  0 
 0  1  0  1  0  0 -1 -1  0 -2  0  0  0  0  0 -1  0  0  1  0 
-1  1 -1  1  0  0 -1  0  0 -1  1  0 -1 -1  1  0  0  0  1  0 
 2  0  0  1 -1  1  0 -1  0 -1  0 -1  0  0  0  0  0  0  1  0 
 0  0  1  1 -1 -1  0 -1  0  0  0  0 -1  0  0  0 -1  1 -1 -1 
 0 -1  1  0  0 -1  0  0 -1 -2  1  0 -1  0  1  0  0  0  0  0 
 1  0 -1 -1  1  0  1 -1  0 -1  0  1  0  1  0  0  0 -1 -1  0 
-1  1 -1 -1 -1  0 -1  1  1  0  0  1  1  0  1 -1  0  0  0  0 
 0 -1  0  1  0  0 -1 -1  2  0  0  0 -1  0  0  0  0 -1  0  0 
 0 -1  0  1  0 -1  0  0  0  2  1  0  0  0  0  0  1 -1  0  0
 0 -1  0  0  0  1 -1  0  1 -1  2  0 -1 -1 -1  0  0  0  0  1 
 0  0  0  0  0  0  1 -1  0 -1  1 -1  0  0  0  2 -1  0  1 -1 
-1  2  0 -1  0  1 -1  0  0 -1  0  0  0  0  1  0  1 -1  0  1 
 0  0 -1 -1  1 -1  0  0  1 -1  0  0  0  1 -1  0  0  1 -1 -1 
-1 -1 -1  1  0  1  1  0 -1  0  1  0  1  0  0  1  0  1  0  1 
 0  1 -1  0  2 -1  1  0 -1  0  0  0 -1  1  0 -1 -1  0  0  1 
 0  0  0  1  1  0 -1  0  0  2 -1  0  0  0  0  0  0 -1  0 -1

Multiplier vector

b[20]=0 0 0 1 1 0 -1 0 0 2 -1 0 0 0 0 0 0 -1 0 -1, norm squared = 10.

Shortest multiplier vector:

b[20]+b[3]+b[4]+b[5]-b[6]+b[11]-b[12]+b[13]+b[14]+b[15]
= 0 0 1 0 0 -1 0 0 0 0 0 0 1 -1 -1 0 1 0 0 -1 
Euclidean norm squared = 7